Two parties are trying to resolve some dispute that involves interest of both. Sometimes it's hard for them to reach an agreement on a fair basis. This is typically when another party who has no interest involved and no bias towards either party comes into play in order to settle the dispute between the two conflicting parties. The term "third party" seems being used quite often. I would also call it a "middle-standing" party or we could say this party is neutral.
These terms seem NOT formal enough to me though. I'd like to see if anybody can come up with a better one.