The first rule before you start resolving these conflicts is to remember that these conflicts are often bigger than you as a person, and you should accept that. The second rule is to never take sides. You will experience the wrath of any stakeholder you will align yourself against, and it won't be beautiful.
Now as a project manager, you must have some negotiation skills (either innate or acquired). So here's what you should do:
The first thing to do is to try to understand what the problem is: why there is a conflict in the first place.
The second thing to do is to identify the impact of the conflict on the project, if the impact is minimal, then let the stakeholders sort it out together, if the impact is heavy, then move on to the next step.
The third thing to do is to find a compromise between the stakeholders, the compromise should satisfy all parties, and you must ensure that no stakeholder feels that another stakeholder "got more" with this compromise. Make sure that your compromise does not affect the interests of other stakeholders.
In case you are unable to resolve the conflicts then alert the affected stakeholders of the impact of this conflict on the project. This will urge them into finding a solution themselves.
Finally, if the stakeholders didn't take any action to resolve the conflict themselves, then throw this whole thing on the project sponsor and let him deal with it himself. He is an a better position than you to do that.