Mediation
Once the participants in a conflict have had a moment to calm down, someone must now mediate a discussion where both sides have an opportunity to understand the other person's perspective. The challenge with mediation is making sure that everyone has a chance to speak without being interrupted, which would inevitably lead to more conflict.
To act as a mediator requires you to be objective, so that you can use logic and reason to help navigate what may be very strong emotions. If you are too closely involved in the source of the conflict, or have too close a relationship with the people involved in the conflict, you may need to find someone else to be the mediator.
It is important to note that mediating a conflict in a professional workplace, and mediating a conflict in someone's personal life are two different things. Generally, workplace conflicts are caused by workplace issues, so someone in the workplace is in a good position to understand the issues, and offer suggestions or even make changes that would help resolve the current conflict, as well as prevent future conflicts.
A personal conflict, as might happen between family members or a married couple, is best left up to professional therapists, with specialized training in mediating that specific type of conflict, which in the case of family members would be family counseling, and in the case of a married couple would be marriage counseling. If you feel that a workplace conflict is resulting from personal issues unrelated to work issues, then mediation is best left up to a professional therapist.
Mediation usually benefits from privacy, such that it is only you and the two parties, as this allows people to speak more freely than if they are being observed by a larger group. Care should be taken to select a location where if people are speaking loudly - often the case at least in the beginning of a mediation session - other people won't be able to hear. This tends to rule out meeting in rooms with thin walls close to where other people sit.
To mediate a discussion, both parties have to speak to one another in a productive manner. This means that they are not calling each other names, are not assigning blame, are not being passive-aggressive or aggressive, are not making unsubstantiated claims, and are not sharing opinions based on nothing more than personal feelings.
Essentially, the job of the mediator in a professional workplace is to facilitate a conversation with both parties, and replace emotion with logic and reason. The goal is to get the two parties to think rationally about their situation, which can be impossible if every time one person speaks, the other person gets triggered to the point where they are so emotional they cannot think clearly. In these situations, it is the job of the mediator to take whatever triggering statement is made, and rephrase it diplomatically such that the other person is open to hearing it.
For example, let's say you were mediating a conflict between two colleagues who have been passive-aggressive towards each other for weeks, until finally it erupted into an all-out conflict where both people were trading personal accusations, blaming the other person for causing problems, and it was clear that neither person had any interest in resolving the conflict on their own. Let's say that both of these colleagues consider you to be their trusted colleague, and so you approach them to ask if they would like for you to act as a mediator to help them talk through the issues they were having with each other.
Assuming they agree, the three of you find a meeting room well away from prying ears, and sit down to have a conversation. You would position yourself such that you can see the body language of both people clearly, as you will need all the information you can get to figure out how the mediation session is going at any given point in time.
Let's say the colleagues' names were Sam and Jessie. You might start by saying, "Sam, Jessie, what seems to be the problem?" At this point both may immediately begin shouting at each other, in which case you will have to intervene to get people to calm down. If you are unable to get them to calm down and have a rational conversation, then you should gracefully end the session by saying something like, "Perhaps now is not the right time to have this conversation, let's give this another try tomorrow."
If you are able to get them to calm down, you might call on each person individually, perhaps by saying, "Sam, what's the issue?" At which point Sam may say, "The issue is that Jessie is always late delivering their work, which is causing me to be late, and I'm tired of getting into trouble for missing my deadlines when it's not my fault." You might then turn to Jessie and say, "Jessie, what's the issue?" to which Jessie might say, "Sam's always complaining about my work being late, but when I ask for help I never get any, and instead Sam just sits around doing nothing waiting for me to finish."
You might then say, "Let me see if I can say back to both of you what seems to be the issue, and please correct me if I've got anything wrong. Sam, you believe Jessie is causing you to be late delivering your work, and Jessie, you believe that you are late in delivering your work to Sam because Sam won't lend a helping hand. Is that correct?"
This process will then need to continue, where you ask questions to help each person to gain a deeper understanding of the underlying issues. Often, the issue can be deeper than anyone expected, so it is important to keep asking questions, and to continue to help people better understand the other person's perspective.
Over the course of the mediation session, people should become calmer, less emotional, and more logical. If the discussion backslides into a conflict, you will need to intervene again, and reset the conversation to the last productive point being discussed. This process will need to continue until all of the underlying issues that caused the conflict are uncovered, and until everyone understands everyone else's perspective, as until this happens, it will be impossible to find a resolution that is satisfactory for everyone involved.