The function descriptions in the stickies here are okay, but don't seem to go into a lot of depth or give many examples. I will attempt to improve them here. Sorry if any of this is repeated from stuff that's been posted before; I just tried to cover everything.
Jung observed two types of general cognition: Perception (taking in information) and Judgment (making decisions/evaluating it.)
He said that there are two styles of Perception: Sensing (which deals with immediate, concrete, tangible sensory impressions) and iNtuition (which deals with abstract patterns and relationships between things or ideas, and other non-tangible information.)
There are also two styles of Judgment: Thinking (which deals with impersonal logic and structure) and Feeling (which deals with personal/interpersonal ethics and morality.)
Furthermore, each of these four processes can be directed inwardly (introverted), which means it's conceptualized subjectively in terms of the self, or externally (extroverted), which means it's conceptualized objectively in terms of not-self.
Introverted attitudes view the outer world in terms of the self's subjective ideal, so they attempt to make the outer world more like the inner self. Their focus is deeper but less expansive--they can see all the implications of one idea at a time.
Extroverted attitudes view the inner world in terms of the non-self's objective ideal, so they attempt to make the inner self more like the outer world. Their focus is broader but more shallow--they can see a wide range of different information at once, but in less detail.
(For example, Ne can see the next few steps down many different paths at once, but Ni sees all the way to the end of one path at a time.)
So, we can derive two kinds of extroverted perception (Pe): Ne and Se (extroverted iNtuition/extroverted Sensing)
And two kinds of introverted perception (Pi): Ni and Si (introverted iNtuition/introverted Sensing)
And two kinds of extroverted judgment (Je): Te and Fe (extroverted Thinking/extroverted Feeling)
And two kinds of introverted judgment (Ji): Ti and Fi (introverted Thinking/introverted Feeling)
Each of these eight functions represents a complete worldview or attitude about life, the self, the outer world the relationship between them. As you learn you will start to see these value systems at work everywhere in everyone all the time, influencing all of their behaviors and ideas so profoundly that most people are entirely unaware of their own biases. As I said, think of them as lenses that color your perceptions/judgments to a far greater extent than you may even realize.
The E/I and P/J letters in Myers-Briggs types are not actually cognitive functions; they simply tell us which directions our functions (S/N and T/F) are oriented and which ones are dominant.
P/J tells us which of our two primary attitudes is extroverted, so if you are xxxP then you prefer Ne/Se with Ti/Fi; if you are xxxJ then you prefer Ni/Si with Te/Fe.
I/E tells us which of those two attitudes is dominant. If you are IxxP your Ti/Fi is dominant and your Ne/Se is secondary (ExxP is the reverse.) If you are IxxJ, your Ni/Si is dominant and Te/Fe is secondary (ExxJ is the reverse.)
So, for dominant/secondary attitudes:
Ne = xNxP
Se = xSxP
Ni = xNxJ
Si = xSxJ
Te = xxTJ
Fe = xxFJ
Ti = xxTP
Fi = xxFP
Now, on to the functional attitudes...first, the extroverted Perception (Pe) attitudes. These are dominant for ExxP types, secondary for IxxP, tertiary for ExxJ and inferior for IxxJ:
Ne, or extroverted iNtuition, is dominant for ENxP, secondary for INxP, tertiary for ESxJ and inferior for ISxJ. It is an outwardly exploratory attitude that encourages us to change, reinvent and experiment with the external world in order to find new and interesting combinations and patterns. Ne looks for novel outcomes and imagines how the things around you could be changed into other, more interesting things. Ne sees new information as part of a larger, emerging, as of yet unseen pattern that extends far beyond the self, and whose meaning will continue to change as the context grows and we discover more of the all-encompassing pattern. Rather than directly confront an issue, Ne will often broaden the context until the issue seems insignificant by comparison to the much bigger and more expansive ideas it imagines.
As with all extroverted functions, Ne needs to be validated by external/objective information to have meaning. So Ne users will often have many ideas very quickly but not know if they're good until they hear other people's reactions to them, or have a chance to experiment and see what happens. Ne wants very badly to be understood and appreciated by others. Note that Ne songwriters (e.g. Brandon Boyd, ENFP) will typically write enough context clues and such into their work that you can put the pieces together and infer what they were thinking when they wrote it. They want others to put the pieces together and get it.
Se, or extroverted Sensing, is dominant for ESxP, secondary for ISxP, tertiary for ENxJ and inferior for INxJ. It is the attitude that what is directly apparent in our immediate physical surroundings is the most important thing to go by. Se leads you to follow your gut instincts, pay very close attention to what's going on around you, and respond to things in the moment in whatever way will make the strongest and most immediate guttural, sensory impact on others. Se users are so present-focused that they're often on the cutting edge of new trends because they place so much emphasis on what is current and new. They like to learn things via a hands-on, figure-it-out-by-experimenting-as-you-go, direct experiential approach (in this way they are similar to Ne) but they are more focused on what is immediately tangible than on what their surroundings might be changed into. They usually pay a lot of attention to their physical appearance and are very good with reading body language and using it to immediately size up a person or a situation and respond instinctively. They can be quite impulsive and prone to overindulgence in sensory pleasures, but they also know how to work a crowd and they tend to make themselves into reflections of current popular trends--whatever will make an impact.
Se is the opposite of Ni because it intentionally focuses on the literal surface meaning of exactly what is going on right in front of you right now, whereas Ni tries to ignore that and see the hidden meaning in what is not directly apparent.
Next, the introverted Perception (Pi) attitudes. These are dominant for IxxJ types, secondary for ExxJ, tertiary for IxxP and inferior for ExxP:
Ni, or introverted iNtuition, is dominant for INxJ, secondary for ENxJ, tertiary for ISxP and inferior for ESxP. It is an inward sense of abstract perceptual shift. Rather than imagine different ways we could change the outside world, Ni acknowledges many different ways we could change the subjective meaning of things to ourselves by looking at them from different angles. Rather than directly confront an issue, Ni will often solve problems by simply looking at them from a different angle. Doing a bunch of community service sucks? Just think of it as an opportunity to get lots of exercise! Note that Ni doesn't think about how to change the outer world the way Ne does; it only thinks about how to change *the way we interpret* the outer world. Ni leads you to try and see "through the smoke and mirrors" to what is REALLY going on below the surface, that other people are not perceptive enough to pick up on...so in its unhealthy form, it turns into conspiracy theories, a la Dale Gribble from King of the Hill.
Strong Ni users like being the person behind the scenes who pulls all the strings (even better if most people don't even realize it) and understands the dynamics of everything on a deeper level than everyone else. They are threatened by the idea that there might be any perspective or angle they cannot see, and as such they sometimes overestimate their own ability to fully grasp and work around the attitudes of others.
As with all introverted functions, Ni doesn't pay attention to external conditions outside the self so it doesn't care if anyone else grasps the ideas the same way the Ni user does. To Ni, I get the significance and that's all that matters. Ni songwriters (e.g. Thom Yorke, INFJ) will often write lyrics that could not possibly make any sense to other people without a direct explanation from the writer, but they don't really care because Ni considers intuition such a personal thing that it can't make its perspective/ideas clear to others very easily at all, and frequently doesn't even bother trying.
For another example, Isaac Newton (INTJ) invented calculus and didn't bother telling anyone about it for 20 years. Ne would have been out showing the idea to others and changing it based on their reactions--but not Ni!
As a result Ne is typically much better at putting its abstract ideas into terms that others will understand than Ni.
On a side note: Ni appreciates definitional freedom (and thus is often annoyed by Ti) in the same way Ne appreciates freedom to change its plan of action abruptly (and thus is often annoyed by Te.) Ti users will tend to frame debates by first assigning precise definitions to terms, but Ni often objects to this by wondering: "How are we unconsciously limiting our understanding by assigning such rigid definitions in the first place?" Ni always seeks to escape the unconscious assumptions that limit its understanding of as many different conceptual viewpoints as possible.
Si, or introverted Sensing, is dominant for ISxJ, secondary for ESxJ, tertiary for INxP and inferior for ENxP. It's related to Se in that it deals with sensory experience, but rather than constantly scan for everything about what's going on now, it relies on internalizing those experiences into an extremely detailed internal map of highly vivid *memories* of those past sensory experiences. This dependence on reliving past experience and using it as a guide for the present leads to an extremely good memory for detail, and a general attitude that going with what we know for sure from having experienced it before is usually best.
Si is the opposite of Ne because rather than relate new information to some larger external, constantly changing pattern, it tries to relate all new information to something it already knows, some sensory data that it's absorbed from its past experiences. This leads to the classic Ne vs. Si battle: Ne wants to try something new just for the sake of doing something different and finding something interesting; Si wants to stick to what we've done before because its vivid memories of direct experience allow us to relate the new information to that past information we've already absorbed.
It's a common misconception that Si users are traditionalists on principle. In my experience, many older Si users (xSxJ types have Si as dom/secondary) are traditionalists because the only source of information they had was their parents and the traditions they were raised with, but these days many younger SJs are much less traditionalist in nature because information is so much more freely available than it was just a few generations ago. Si is not into tradition just for the sake of tradition; it just likes to relate new information to something it already knows. Rules and traditions can be a convenient way to do this, but it's a mistake to believe that Si always leads to traditionalism for its own sake.
Si also does some really cool stuff like perfect pitch...I have one ISFJ friend (Si dominant) who can tap into his past sensations of what a particular note sounded like and use it to identify some note he hears now as a G#. That's amazing to me...as an Ne dom I only understand notes in terms of their relationship to other notes in a larger pattern; Josh just taps right into his detailed sensory memory and can identify the note by remembering what it sounded like before, on its own.
Si doms like to collect objects and facts that evoke pleasurable memories from the past. An Si who's into history will collect books, photos, stamps, etc...an Si who's into music will collect instruments, sheet music, photos of concerts, and so on. Many Si types love scrapbooking because looking back at those old photos evokes those powerful, highly detailed sensory experiences from the past.
On a more morbid note, Dexter (from the TV series "Dexter") is probably ISTJ. He collects blood slides because they evoke the detailed memories of his most enjoyable murder experiences from the past. =/
Next, the extroverted Judgment (Je) attitudes. These are dominant for ExxJ types, secondary for IxxJ, tertiary for ExxP and inferior for IxxP:
Te, or extroverted Thinking, is dominant for ExTJ, secondary for IxTJ, tertiary for ExFP and inferior for IxFP.
It's an attitude that encourages an external, objective standard when dealing with logic, impersonal facts and ideas. Te, when arguing, will tend to cite appeals to authority and other widely accepted, externally focused evidence; i.e., citing books or prominent authors/studies, or any widely accepted consensus among the external world of people who study the topic in question. "The experts all agree that this is the case" is a very Te-oriented argument, because it relies on external standards and context for its evaluation of logical decisions.
For this reason Te people will usually insist on seeing quantifiable, repeatedly demonstrable, empirical evidence before accepting anything. If you can't put it in a test tube, measure it and repeat these results any time for all to see, it's not valid. The scientific method is extremely Te-oriented. From the Te perspective, there is no such thing as logic without this sort of externalized validity, because impersonal ideas are to be shared and agreed upon by large groups instead of individuals (the same way Fe treats ethics) and determined by objective consensus. Te users tend to find Ti selfish and unyielding in its insistence on fitting things into its own personal logical framework before accepting them, rather than taking widely accepted external evidence or consensus seriously.
Strong Te users are efficiency experts. They are typically very good at translating a theoretical idea into a fluid, external process that gets effective, measurable results that can be repeated and verified on schedule. They usually do very well in management positions that allow them to focus on process over theory in order to maximize efficiency and bring about the desired goal while expending the smallest possible amount of resources (especially time.)
Fe, or extroverted Feeling is dominant for ExFJ, secondary for IxFJ, tertiary for ExTP and inferior for IxTP. It is an attitude that encourages adherence to the ethics of the cultural/social/familial groups we feel emotionally connected to. Fe leads you to derive your moral viewpoints from some sort of externalized consensus. This doesn't mean you automatically fall in line with whatever moral viewpoints happen to surround you, just that (unlike the accompanying Ti view on logic as something you don't need external input to understand) you don't see how ethics can be decided reasonably without some sort of external context. (Fe views ethics as dependent upon collective consensus in the same way Te views logic/impersonal ideas.)
Fe leads people to adjust, hide or set aside entirely their own emotions in favor of fitting the emotional needs of the broader groups that are important to them. This leads to a certain respect for the common consensus among those important groups regarding interpersonal behavior and treatment of others. If you were to criticize someone's behavior from an Fe standpoint, it would be from the standpoint of, "Your behavior is inconsistent with the group's standards--most people would consider it wrong or inappropriate." Fe appeals to the collective morality of the whole; the fact that "most people would agree" serves as externally objective evidence to support Fe's moral standpoints.
People with strong Fe are typically good at saying just the right thing that fits in with the moral expectations of the audience. For this reason Fe tends to make great politicans because strong Fe users often make outstanding, charismatic public speakers who can play off the emotions of others to rally groups toward the desired cause. They are excellent at organizing, leading and delegating tasks to others with an interpersonal style that gets the job done while still appearing socially appropriate and respecting the emotional needs of others (so long as those needs are reasonable within the group's objective framework of ethics.) They understand how to perform the social/cultural responsibilities expected of them and they expect others to do the same, and if you're not fulfilling these responsibilities they're very good at appealing to the crowd to deliberately make you look like an asshole in front of everyone. ("Look everyone, this guy doesn't fit with our collective moral ideals!")
Fe considers it paramount to show overt displays of loyalty to the people in the groups it feels connected to, which includes helping out friends/family whenever possible and receiving similar displays of loyalty in return. (If these displays are not reciprocated Fe may take this as a sign that the other person is not loyal.) Fe tends to see Fi users as selfish for refusing to adapt their feelings to the feelings of others in service of the good of the larger group, and for ignoring objective standards on ethics in favor of purely personal ones.
The whole idea behind Ms. Manners is very Fe--Fi would wonder why anyone cares about any external consensus on ethics, because to Fi ethics are purely subjective. Fe is concerned with adjusting to the ethical standards as established objectively by the groups it feels are important.
And finally, the introverted Judgment (Ji) attitudes. These are dominant for IxxP types, secondary for ExxP, tertiary for IxxJ and inferior for ExxJ:
Ti, or introverted Thinking, is dominant for IxTP, secondary for ExTP, tertiary for IxFJ and inferior for ExFJ.
It's an attitude that encourages subjective logical decision-making based on our personal and directly experiential ideas of what fits into an impersonal logic framework and what doesn't. When it comes to logic/impersonal ideas, Ti reasons, external consensus can go to hell because it might very well be wrong, no matter how many people believe it or how many experts claim to know the truth. Ti seeks truth for its own sake; it wants to understand the relationships that force frameworks of information to fit together into cohesive wholes. Ti is focused on the blueprint, the design, the idea--while Te is focused on the application of that idea into an objectively measurable process. Externally measurable application is not nearly as important to Ti as internal structural integrity and logical consistency with itself.
Ti appreciates structural symmetry, balance, and the beauty of symmetrical models that elegantly explain and organize real world phenomena (perceived by Ne or Se) into neatly arranged categories. Ti people are usually very good with pure logic in a vacuum, as Ti simply "knows" inherently what is logical and what is not, and will defend this sense of logic to the death just to prove a point. Te people, on the other hand, are more concerned with what tangible USE can come from an argument--which is often none. This is why INTPs will argue hypotheticals all day but INTJs will rarely bother trying to convince you. For the INTJ, Te simply doesn't see what useful goal would be served by trying to change your opinion.
Ti: What logical relationships necessitate this system working the way it does, and how can I make them make sense to me?
Te: What externally verifiable, quantifiable evidence can we show that this is logical, and what tangible goal can be served by spending our time on it?
Another good example is music theory...when I learned chord theory I naturally wanted to learn the rules of how chords fit together so that I'd understand the entire system holistically and could theoretically figure out any chord. Ti likes to figure out entire systems just for the sake of getting a glimpse of complete truth ("I want to play guitar, so I will learn the system of rules for how chords are built so I understand the whole thing at once"), whereas Te is much more goal-oriented and always wants to know how this system can be applied to something externally useful or used to accomplish our predetermined goals...so Te would be more inclined to first figure out what the goal is ("What do I plan to use my guitar playing for?") and then learn only what's necessary to complete that goal. ("I want to learn 'Freebird', so I will learn the chords and techniques necessary to play that song.")
Te takes a step by step, sequential and linear approach based on which steps are needed to complete its goals, while Ti tries to understand the entire system as one big unit simply because it's interesting and stimulating.
I borrowed the following baseball analogy from Lenore Thomson, because it works so well:
So let's say you're playing baseball. Te would tell us that if the runner doesn't reach the base before the ball gets there, he's out--period. That's objectively verifiable and can be shown logically, through external empirical evidence/consensus of experts that it is always the case, and it can be quantified and measured precisely without any personal emotions getting involved. (You can see why so many TJs are research scientists, especially NTJs.)
But Te won't do us any good when we're the runner trying to decide whether to steal 2nd base or wait for another hit. The situational logic in this case is subjective Ti because it requires us to reason out what makes sense at the moment according to our direct experience--the logic at play here cannot be precisely quantified in an externally verifiable manner.
Fi, or introverted Feeling, is dominant for IxFP, secondary for ExFP, tertiary for IxTJ and inferior for ExTJ. Unlike Fe, Fi leads you to draw ethics purely from an internal, subjective source and finds Fe's collective approach to morality shallow and fake. Since ethics are purely a personal ideal in Fi's view, all personal feelings are sacred and allowing any outside views to affect them is patently unethical. Fi treats ethics in the same way Ti treats logic, in that it's something that requires no external context to understand and that should not be influenced or changed by any outside forces.
The ability to express one's personal feelings and inner self freely and maintain a strong sense of personal uniqueness and individuality is of utmost importance to Fi. (It's also important to Ti, but for different reasons and in different contexts.) I have found that many Fi users dislike typology in general because they feel that "putting people into boxes" suppresses their sense of personal identity, and that people are too unique to be categorized so easily.
Fi people are typically very good at picking up emotional vibes in the tone of voice and word choice of others. They often know what you're feeling even better than you do because they're aware of the subtle effects that different emotional states have on our behaviors and can pick up cues about how you're feeling that you didn't even intend to give off. For this reason strong Fi users are profoundly empathetic and tend to understand and identify with basic human needs on a profound level. If you were to criticize someone's behavior from an Fi standpoint, it would be in the form of, "What you are doing is hurting my feelings/violating my ethics/preventing me from fulfilling my basic human needs." Strong Fi users often feel a certain connection to the beauty of nature, animals, and life itself, simply "knowing" deep inside themselves that life is sacred and all individuals possess inherent value.
A mature Fi user is extremely in tune with the emotional needs of others and very supportive of and responsive to them. An immature Fi user is overly preoccupied with his own emotional needs and will act passive aggressively toward people who don't bend over backwards to cater to how he feels. Note that both Fe and Fi users often feel a strong sense of moral obligation to their loved ones; the difference is simply the source of this obligation. If it comes from an external/objective cultural standard, it's probably Fe--if it comes from a personal sense of moral responsibility that deliberately blocks out external influence, it's probably Fi.
Fi doms are ethical perfectionists in the same way Ti doms are logical perfectionists. They seek a sense of internal balance and harmony with their surroundings that feels right in their own individual way. Note that introverted judgment (Ti/Fi) seeks depth and specificity while extroverted judgment (Te/Fe) seeks broad applicability. Ti wants to define exactly what is logically correct under an extremely specific set of circumstances that may never happen in the real world, while Te seeks widely applicable objective consensus that can apply in many different situations. Fi seeks to determine precisely what the user feels is morally right regardless of external application, while Fe seeks widely applicable, generalized ethical rules that can serve to govern entire groups. Ti/Fe = I think/We feel; Fi/Te = I feel/We think.