Before attempting to mediate disagreements, make sure you look at all sides of the argument. While you shouldn't take sides when attempting to mediate, you don't want anyone to think you're overlooking their point of view. Take some time to think about why everyone feels wronged or hurt in the situation.
You will have to resist the urge to judge here. It can be difficult to look at family situations objectively, especially if you're emotionally involved in the conflict. Try to think about how you would react hearing about the conflict as an outsider. Examine why both parties feel wronged and hurt. Consider your family's history and what role, if any, this plays in the drama. Also, look within yourself. Have you done or said anything that could have escalated the situation? How do you think other family member's felt about some of your actions? Try acknowledging this out loud, such as by saying, “I shouldn’t have raised my voice. I am sure it was upsetting for you to hear that.” Acknowledging this out loud and leading by example can set the tone for other people to follow suit! You want to behave admirably and have people want to replicate that.