We are always facing conflicts of one kind or another. This is life. Actually, all of life is a matter of resolving conflicts; the universe was created as a place of challenge. We are put here with the illusion of being separate from the Infinite, and our job is to break down that illusion.
Similarly, the body and soul are in constant conflict, and our job is to harness the energy of the body for the work of the soul. We have a desire to eat to satisfy our taste buds, yet our mind wants to eat to sustain good health. The soul wants good health in order to be vibrant and sharp in service of the Almighty. Just as it is work to train the body to eat healthy, it's work to train the mind to think of our higher purpose.
Kabbalah teaches that the universe is built on a tripod, consisting of two opposites and a harmonious resolution. This is the pattern for everything in life. Stability and confidence is impossible without the support of all three legs of the tripod. We cannot have "shalom" without all three. This tripod - and its attendant challenge of conflict resolution - is inherent inside each one of us, in all of nature, of mankind as a whole, and in the spiritual realm as well.
ASSUME THE BEST
Moses had to deal with many rebellions. It would be easy for him to have fallen into a pattern of knee-jerk reactions to these rebellions, but Moses took the time to assess each problem and try to work out a reasonable solution. He didn't assert his authority, but tried to reason with the complainers. Only with Korach and his followers did Moses resort to a showdown, because the whole issue with Korach was a challenge to Moses' (and God's) authority.
Just as Moses went out of his way to try to reason with complainers, we too should always give people the benefit of the doubt before resorting to drastic measures. While there are patterns to look out for, we do need to look at each conflict on its own in order to find the best solution.
On the other hand, we need to be on the lookout for someone who is an argument waiting to happen. No amount of discussion will help in that situation. It's a waste of time and effort.
LOOK AT THE OTHER SIDE
It's so easy to ridicule the other side in an argument. It's harder to try to look at things from their point of view. Once we're entrenched in an opinion, human nature makes it very difficult for us to change our mind. We are invested. Our ego is on the line.
A heated argument can instantly change to a reasoned discussion when one person says to the other, "I never thought about it that way. You have a point. I'll have to think about it and get back to you."
Human nature being the way it is, you may not find it easy to do this. However, at least think through the issue through the other person's eyes. Try to understand their point of view. The two of you might just be holding the same view from two different angles. A diamond has different facets. Sometimes an issue is like a diamond. I once witnessed a disagreement between two people. Sally said that a certain person was blind, and Jack said, "No, he's not. He has dark hair." The ridiculous argument went on for a minute before the resolution became clear.
THREE FINGERS POINT BACK
When you point a finger at someone else, three fingers point back at you. Often the very thing that bugs us about our spouse, child, or friend, is something that you have deep inside you and it bothers you. A Talmudic statement says, "All who come to invalidate others, invalidates with his own blemish" (Kiddushin 70a). Is it psychology? Is it a spiritual challenge? I don't know, but it's very common.
A word of caution: If you find the finger pointed at you, don't tell the other person they have the problem they're labeling you with. They won't see it. But ask yourself if that's the case. At least if you understand this dynamic, it makes it easier for you to deal with the situation.
IS IT WORTH IT?
Often, the best way to win an argument is to lose it. This applies in a case when the argument isn't worth the fight or the hard feelings. You can give in and say, "I guess you're right." It won't kill you, I promise. (Assuming we're not talking about crucial life issues.)
Every so often there's a news report of a murder in a bar over something like which football team is better. This is an extreme example but we have similar situations in our lives where we let bad feelings ruin a good day or a good relationship over something minor. Even big issues like global warming or foreign policy, since the discussion of two individuals over a back fence probably won't change major issues anyway, why not give in?
Truth is not always more important than harmony.
Find a disagreement in your life, and resolve it using one of the above solutions.