GM stands for Game Master, and is usually used for games with large company-run servers such as MMOs, as GMs are usually employees of the company. GMs are responsible for helping players when they get stuck with some kind of bug, or are being harassed by other players. They are also responsible for enforcing the rules of the game, and occasionally as passers-on of news and information regarding the game (such as news of a server going down for maintenance).
Some tabletop games refer to the person in charge of running the storyline of a game the Game Master. Game Master is the equivalent to Storyteller (Vampire: The Masquerade) or Dungeon Master (Dungeons and Dragon).