In general terms, a mediator is a neutral party that assists in negotiations and conflict resolution, the process being known as mediation. One such mediator was Riva of the planetRamatis III. (TNG: "Loud As A Whisper")
Given that the dispute was expected to end in war if not resolved, and Tellarites and Andorians together joined the Coalition of Planets a few months later, it seems likely that they succeeded.
On the planet Rubicun III, the title Mediator was given to specially designated Edo who were charged with the administering of punishment for any crimes committed in the punishment zones. The Mediators arrived on the scene of a crime usually within minutes of the transgression. Pending admission of guilt or witnesses to the crime the Mediators would administer punishment – always execution – almost immediately. Execution was carried out by injection of a fast and painless poison. (TNG: "Justice")