Resolve Dispute is a quest you can be given by your minister, your spouse or a lord's spouse in order to persuade two lords to end a conflict between themselves. Depending on who gives you this quest, you have to complete it in different ways.
Given by another lord's spouse[]
When given by another lord's spouse, they will request furs to make a velvet robe to please said lord. After that, you may speak with the lord and complete the quest.
Given by your minister or spouse[]
When you are king and/or married, your minister and/or spouse will occasionally inform you (upon asking for their advice to strengthen your kingdom's unity or improving your standing in your realm) about a dispute that is happening between two lords. You are then given the chance to solve this problem by persuading each of them to resolve their dispute. Note: you are given this quest more often when you're a vassal.
When talking to one of the two lords, you can agree with him, as to take his side in the argument. This will resolve the dispute and boost your relationship with the lord you agreed with, but the other one may or may not take offense depending on his personality and your relationship with him. If you're a king, this decision will also be viewed as controversial by your other vassals.
Alternatively, you won't risk offending either party if you ask both lords to "settle their differences". This is, however, more difficult to achieve, as it also requires a good enough persuasion skill. When successful, this action will give you 3 honor, 500 experience and 25 renown.
If you're a king, denying to resolve a dispute after you've accepted to do so will provoke a negative reaction from all vassals who don't have a good relationship with you. If you fail the quest by running out of time, your right to rule will go down by 2.