I have always maintained that solving conflicts or disagreements amicably is one the practical ways through which you can put an end to any type of conflict you are having with people around you. Conflict in the workplace can get you into serious trouble if you don't go about it peacefully.
The first thing to do is to look for ways to end this disagreement with the other person. Try to meet with the other person or party, and you need to be sure of a perfect time for this. Let it be clear to the other person why you think the meeting is necessary. Start by giving the other person a good time to tell their own side of the story.
Try to listen to him without reacting or interrupting him(her). Go back to the issue that led to the conflict. Both of you need to check your areas of agreement and disagreement. Let the person see why you need to reach a balance on it. Talk more on this and ensure that both of you reach a solid point.