Wooh, I have been watching this show over and over again since the day it aired first, so move out of my way people lol...
Aang (ENFP 9) - like others said, classic child ENFP...probably a 9 enneagram. This is easily seen with his struggle to give up his fun and accept responsibility like many ENFP children. He just has that charm that everyone can fall in love with...and he tends to "talk" through his hurt with people and dream out loud.
Katara (ENFJ 2w1) - While she does use Ni and seems more introverted her primary function used is without a doubt Fe...everything for her revolves around serving the people around her. Wherever there is a need for help...she will fix the problem. Every ENFJ I've ever met has been like this lol. Especially the mellow ones...and they lash out like she does too :P. INFJs just have a completely different aura about them.
Sokka (ENTP 8) - Sokkas probably the iffiest character to me just because he has so many levels to him. While he does hold his values in a way an SJ might...he only gets like that when he's in a panic. His dominant function is definitely Ne though...I mean, his creativity is what people value highest in him after all.
Toph (ISTP 8) - This chick is just as ISTP as it gets guys :). She does not like to be controlled but does not make a huge point about it like an ESTP does. Shes brutal...but boy is she awesome :)
Zuko (INFP 4) - okay...don't kill me but my father and I talked this over for hours. While he does seem rather INTJ at times you see that he is clearly not when observing his childhood and how gullible he is to Azula. I mean, would an INTJ, even unhealthy, believe her so easily after years of torment? Nope, an extremely hopeful INFP would though. Plus he doesn't stick to his ideas, he sways between like an INFP (this destiny, that destiny, bad, good). He also seems ISTJ, but this to me is a struggle to be accepted and loved in the freak culture he grew up in, one of the biggest struggles INFPs face..plus when hes angry he emotionally lashes out much like INFPs yet again...using horrible logic :P. His goal is to be loved...most INTJs wouldn't give a care and would also be better planners at catching Aang :P. There, done :D
Uncle (INFJ 1) - He has mastered the use of his Fe, yes, but Ni is what rules the man. Everything he relies on comes from spiritual type things and he is always happy to lead others the right way. And he's just got cheesy INFJ old guy humor that my father has :P.
Azula (ENTJ 8) - Severely bonkers ENTJ...shes like a crazy control freak ESTJ, only with the ability to control people more effectively. God, she scares me. She is also focused highly on being "the best". This is seen with how she freaks when her mother doesn't view her as highly as zuko...even though she's better at everything else...
Tai Li (or however it is...ESFP 7) - the difference between an ENFP and an ESFP is not one is smart and the other is dumb...its the physical vs the mental. Tai Li is more in touch with her physical side, seeing as she is a master at controlling her body and she aims at perfecting it with the whole aura thing.
Mai (INTJ 6) - Most would say ISTJ but shes not so strictly bound to the fire nation system. Shes sees what happens before it takes place and effectively goes with the situation the best she can to avoid problems. The whole boredom thing probably comes from her being able to do this. Plus she hates Azula, and clearly isn't afraid of her. She just chooses not to show it in front of her to avoid something she sees as unecessary.
Ozai (ESTJ...) Oh...uncles son died. I should take the throne you know. Oh, I have to kill my son because of that comment? Okay? A very power hungry unhealthy ESTJ who hates anything that appears as weak.
Suki (ISFJ) - Very strict sense of duty to the Kyoshi warriors, but not to the point shes cold. Very loyal...
Okay, from here I'm just labelin' away without descriptions because I'm tired. I don't remember a lot of names, but I remember descriptions lol...
Gran Gran - ISTJ
Momo - ESFP
Appa - INTP
Monk Kyatso - INFJ...love this guy :D
Zukos mother - ENFJ...very fierce ENFJ
Z's mother's dad - ISTJ
Roku - INFJ
Earth bender prisoner son dude - ISFP
Jet - ESTP
Longshot - ISFP
Smellabee - IXFJ maybe...
Yue - INFP definitely
Waterbending master teacher - ISTP
Zhao - ESTP
Earth King - ISFP
The boulder - crazy
Sword teacher - INTP
King Bumi - Awesome ENTP
Dai Li Leader - INTJ...argue with me and I will give reason hahaha
Hu (swamp guy) - ISFP
Jong Jong? was that the first firebending master - INTJ
Little boy Zuko stayed with while being a refugee - ESFJ
FREAKY SMILE LADY - lots of personalities...but the mask is ESFJ
Sand Library scholar - INTP
Lizard lady who tracks stuff - ISTP
Crazy guy selling two headed fish at the polluted place - ENTP
Mean sun warrior (time for them to be eaten) - ESTP
Smart Sun Warrior - ENFJ
Guru - INFJ...he radiates that INFJness
I know I forgot people...but I tried lol...