In Starcraft 2, a Referee is a player who can watch a game, talk to players (usually about rules), change game speed, pause the game or kick players. It is essentially an enhanced Observer player(who can only watch a game).
I found something about a Referee's role on the MLG website, here are some bits of it:
All interface changes must be supervised by a Starcraft 2 Referee or the Starcraft 2 Tournament Director.
If a Game is started without the approval of a Referee it will be restarted.
Players may be disqualified if they leave their Station without their Referee’s permission, or are otherwise unable to play. Referees may set a time limit for a Player who has requested that they be allowed to leave their Station, however, Referees may also deny a Player’s request to leave their Station. Players may be disqualified if they haven’t returned by the end of the Referee’s set time limit.
In order to dispute Game results, Players must notify their Referee that they would like to Protest the Game. In order to dispute Match results, Players must notify their Referee that they would like to Protest the Match. Players must notify their Referee of their intent to Protest a Game before a new Game has begun. Players must notify their Referee of their intent to Protest a Match before the Match’s Score Sheet has been submitted to the Tournament Director.
No Pausing a Game without Referee’s permission. If an issue arises that requires a Pause, Players should contact their Referee immediately. Referees may approve or deny a Player’s request to Pause the Game. Pausing a Game without Referee’s permission will result in a Warning. If a Player receives a 2nd Warning they will Forfeit their current Game (See Gameplay Rule #17).
Referees will conduct a Blind Pick of Races for Game 1 of a Match.
Players who quit out of a Game before it has ended, without Referee’s permission, will lose the Game.
Only MLG Staff members may connect to the server as an observer. Inviting someone to observe a Match without Referee’s permission will result in a Warning. If a Player receives a 2nd Warning they will Forfeit their current Game (See Gameplay Rule #2).
These rules are of course for human referees during the tournament, but we may as well assume that an in-game Referee has those same duties during a game (eg: the only way to kick a player from a game is to actually be in game, or unplug his computer but I don't think that's allowed in MLG games)