With people I don't know/don't care about I tend to avoid conflict as much as possible. I try to smooth things over, avoid contact, smile and nod to appease them and then do whatever I want when they aren't there.
With people I do care about and am close to I wouldn't say I'm confrontational, I still try to avoid conflict wherever I can, but I also won't let small tensions build up. If I sense something is off between me and someone I care about I will try to bring it up tactfully and seek openly seek resolution, rather than trying to solve it by guess-work or brood over it or wait for it to pass.
When I do get into a situation of conflict with someone I care about I tend to cling to them, often even physically, untill the conflict is resolved. I don't leave, and I don't like to let them leave before we are 'okay' again. Even if I'm really mad and yelling at someone, I will often grab onto part of them if they try to move away from me because I want it to be clear that the conflict isn't going to affect my care for or commitment to them.
I usually try to understand someone else's perspective when we're having some kind of conflict, and I seek to explain my perspective to them. I feel that this is the only way to really resolve something. You can't just make someone agree with you, you have to both seek understanding and reconciliation. I usually don't want them to just tell me 'okay you're right' and I don't want them to expect me to do that either. I like to talk things through and for there to be both give and take. I also like to do everything we can to make both people's ideals/preferences possible. And I'm a big fan of agreeing to disagree, I feel that is an important sign of respect and maturity.