As a project manager, you should have some excellent negotiation skills, and these negotiation skills should be put at use when managing stakeholder conflicts.
If two ore more stakeholders are having conflicts together, then you need to resolve these conflicts through negotiation: you need to find a common ground between the conflicting parties, and you will need to ensure that you're not taking the side of any stakeholder, regardless of whether he is right or not. Because if you start taking sides, you will lose your status as a neutral party trying to resolve the conflict. You need to propose a deal and make it seem like everyone won something from this deal.
Now if you're having a conflict with a stakeholder yourself, then the first thing to do is to ask yourself whether this conflict is personal or professional, you should be honest to yourself in answering this question.
- If it's personal then you should talk to your project sponsor on the best way to resolve this conflict, because emotions will be definitely involved if you try to resolve the conflict with the stakeholder yourself (like face-to-face), and you don't want that.
- If the conflict is professional then the best thing to do is to meet with the stakeholder and try to find a common ground between the project (not you) and him. Always remember that you must be humble with the stakeholder (he'll appreciate that).
You can't leave a stakeholder conflict unattended, because it will simply backlash on you, and you will never get the project done when there is are stakeholder conflicts that are not addressed.
Quick note: Always remember that each and every stakeholder has an agenda.