In my experience there are 4 kinds of conflicts:
1. Fights for limited resources. Goodness knows how many wars have been started over this one!
2. Symmetrical Misunderstandings. The kind where both parties feel wronged and are unable to communicate their grievances to the other party in a productive manner. These I think are the best kind of conflicts and the ones that many optimists like to think all fights amount to. They're the kind that can be solved through mediation and ultimately compromise. I hope I'm never squabbling over resources, but certainly this is a kind of conflict I've experienced and thankfully been helped to solve.
3. Asymmetrical Fights. This when one party has a big problem with the other and the other party's only complaint is that they want to be left alone. I remember in 4th grade I had a classmate who hated me. She came from an abusive household and as much as I can figure it out, I think she just hated me because I represented everything she didn't have. I was nice to her, because, especially back then, I was nice to everyone. She did things like push me down stairs, spread rumors about me and at one point try to crush my foot under a heavy wooden chair. The teachers kept trying to mediate us, but there was no way to mediate a conflict where she wanted me obliterated and I just wanted to be friends. Not fun, but a real learning experience I must say. I still try to be nice, but she's the one who taught me that people can hate me for nothing I did to them and that my best course of action is to prove that I'm not worth picking on.
4. Unreconcilable Differences. When you both have conflicting opinions that can't be resolved. This is the one I hate the most, because there's not necessarily a value judgement involved, you're just not going to agree unless something fundamental about your personalities shift. The interesting thing about it is that it requires a critical mass of disparities. Say, you're an atheist and the other person is devote monotheist, OK, you can still be friends, just leave religion out of it. Now it turns out the other person is a passionate political vegan and you eat anything that moves and the less you know about its origins the better. Alright...don't discuss religion and don't go out to eat together ever. Now it turns out the other person is a big believer in polyamourism, pansexuality and recreational sex and you're a strict believer in waiting until marriage to bond with another person that closely...eventually you get so many fundamental beliefs that are different that there is just no way to interact with the other person without fighting. Thankfully it's also the conflict I see come up the least and I'm proud to say I have many friends who have wildly different fundamental beliefs than me and that's kind of why I find them so interesting. You have to get a pretty big stack before that other person is just impossible to interact with!