Often in conflict there are three sides, your side, their side and the intersection. This is demonstrated in the figure below. During conflict situations there are things that both parities agree upon, the intersection, and the remaining things are usually the underlying problems of the conflict. Seeing the problem from the others point of view and working on issues one at a time enable resolution of the problem. As you seek first to understand and then be understood you should also work on actively listening, not just passively hearing.
Conflict can escalate when one tries to talk more than listen.
If it took a long time for the conflict to begin, it will take time to end it. Work on one small issue rather than many large ones. Almost always, it is a laughably small wound that causes the first hurt in a relationship. Focus on the issues not the individuals. The problem is the problem.