It's always hard to watch two people you love have an argument. When two mutual friends are fighting, the threat of a rift between them is both personally painful and socially awkward. For this reason, it's easy to want to do something to help them sort things out and they may find that you are the perfect neutral party to help them do so. To avoid making matters worse, proceed with caution and know when to step back.
Stay out of it unless both parties want you to mediate. Let them know that you'd be willing to see if you can help them smooth things over, but if they don't seem receptive, let it go. Your involvement in this situation can only be helpful if both parties trust you and are interested in reconciling.
State firmly that you don't intend to take sides. Inform you friends that you will remove yourself from this situation if they push for you to take sides or agree with statements you feel are unfair about the other.
Accept that you may not be able to rectify the situation. Let go of any demands you might be placing on yourself to fix the rift, and resign yourself to only doing what you can to improve understanding and assist your friends' own process of reconciling.
Get in touch with how you feel about the situation. Consider excusing yourself from trying to help if you feel particularly angry at one or both parties or feel frustrated that you're in the middle of the situation. Try to process your emotions about the situation and come to a place of neutrality and compassion for both parties before getting involved.
Listen to each party. Avoid trying to push them to understand the other party at this stage, and avoid passing judgment on any actions. Empathize with each party's feelings about the situation.
Identify communication difficulties. Look for assumptions both parties have made about the other's intentions and motivations that you believe or know to be false. Encourage each party to question what they think about these things.
Speak to both parties a second time. Make statements about what you think the misunderstandings at play may be. Explain the other party's point of view and motivations in the most neutral and factual terms you can. Let both of your friends know that you would personally appreciate it if they made an effort to understand and to make up for your sake.
Let your friends speak to each other and reconcile in their own time. Accept that you've done what you can to help the situation improve, but that the choice to make up ultimately belongs to them, along with the process of how they do it.