A mediator is an impartial person who helps reconcile a dispute between two or more parties. A moderator is someone who presides over a discussion. Some dictionaries list the words as variants of each other in a few senses, but the words are generally confined to their separate uses in edited writing.
Both have corresponding verbs—mediate and moderate. To mediate is to help resolve a dispute, and to moderate is to guide a discussion.
Germany acted as a mediator between Israel and Hamas in an intense round of negotiations last year which came close to bearing fruit. [Guardian]
Moderator John King moved on to Ron Paul, staying focused on the subject of jobs and the economy. [Politico]
Negotiations are being conducted under the supervision of a federal mediator, and are continuing. [Los Angeles Times]
The atmosphere between the two men when they met for a televised debate on the dispute was described by its moderator as venomous. [The Australian]