Conflict Resolution. I could write a book on that subject by itself. Here are a few suggestions to make resolving conflict easier:
1) Try to "walk a mile in someone's moccasins." Imagine the other person's viewpoint. Why do they have the opinion they have, and what in their life could be causing them to act the way they are? Having empathy and compassion for others can frequently dissipate your own anger and irritation with them.
2) Resolve conflicts at a time when you can both be calm and reasonable. If a discussion becomes too heated, postpone it. "We're both becoming too angry to be reasonable about this right now. Can we discuss it when we get home from work tonight?"
3) Use statements like, "When you do X, I feel Y." "When you leave dirty dishes in the sink it makes me angry because I have to take the time to wash them." Only speak about your own feelings and your own experience. Avoid saying things like, "You always do X," or "You don't care about Y." Stick with your own feelings.
4) Really listen to the other person's opinion/complaint/position, and then summarize what they've said before presenting your own. "I understand that you feel you don't always have time to wash your dishes. I become irritated when you leave them for me, because I don't always have the time to wash them, either."
5) Have the other person come up with the solution, or ask how you can resolve the conflict together. When someone creates their own solution, they're more likely to stick to it. "The dishes really need to be done, and both of us have busy schedules. What do you think is a good way to make that happen?" or, "The dishes really need to be done, and both of us have busy schedules. How can we work together to get them done?" Always be willing to compromise.
6) In a situation where no middle ground can be reached - such as differences of opinion - agree to disagree. Others are allowed to have their own opinions, just as you are. "I understand how strongly you feel about Candidate X, but I really like Candidate Y. I guess we have different opinions! May the best man win."
7) When facing criticism, evaluate whether or not it's true. If it is, agree. "I didn't realize how often I crack my knuckles and how much it annoys you! I'll pay more attention in the future." If it's not true, diffuse it with a statement such as, "I had no idea I crack my knuckles. You've given me something to think about," or, "Sounds like you think I'm working too slowly. I'm glad you let me know."
Conflict Resolution is a skill that can take time to develop. If you feel you've made a mistake, go back to the other person and apologize - "I got a little too hot under the collar before. I'm sorry, and I'd like to revisit this discussion when we're both calmer."