What are some ways to resolve conflicts without fighting?
That's a really good question, and I'm glad that you asked it! There are lots of ways to resolve conflict without getting physical.
Obviously, the place to begin is talking it out. When discussing the issue, it seems that using "I" statements instead of throwing blame at the other person helps to keep things cooler. So, for instance, if they let you down in some way or betrayed your trust, tell them that "I expected you to keep my trust and I felt hurt and betrayed when it was broken", instead of saying "You betrayed me and you let me down." The "You" statement will put them on the defensive and things are less likely to get solved if someone's busy trying to defend themselves instead of finding a solution or compromise.
Don't attack the person, attack the situation. Instead of name-calling or bringing up past situations that are irrelevant, focus on the issue at hand. Once name-calling starts going on, that's only going to push you farther away from a resolution.
Let them know what you expect now. Do you expect this to never happen again? Do you expect them to apologize to you? Listen to their expectations as well.
If you can't seem to talk to each other without arguing and getting angry, maybe writing a letter, stating your position and how you are feeling will help at least to ensure that you were heard. Ask the person to write you back and explain what they are feeling. Then maybe you'll be able to talk, once the main issues and points were touched on in the letter.
If you feel that things were not resolved between the two of you, or if they get too heated and you can't solve anything, call in a neutral third party. Maybe your school has a peer mediation program? Request a guidance counselor, or teacher that you both trust, to help you out.
If the conflict is with someone that is not in your school, say if it's a family member, you could still check with someone at school to see if they will help, or ask someone to recommend a counselor that could help you get things sorted out.
I hope I've been able to help you. Good luck.