GMPC

Sometimes, a Game Master doesn't have enough players to run a decent game. Or else, he just wants to have his cake and eat it too. Or perhaps the campaign needs a Mr. Exposition and the rest of the party doesn't fit the bill. Either way, he starts out an important NPC to travel with the party and fill any missing roles no one else wants to play. It's almost like the GM has a Player Character of his own, thus this concept has come to be known as the GMPC.

Admittedly, this kind of character can be done well - adding flavor to the campaign and immersing the players in the world the Game Master has set out to create. This kind of GMPC often serves as a Non Action Guy in order to avoid stealing the spotlight from the players, or at least some form of White Mage. (passive GMPCs often end up in the 'healing' role)

However, it's all too easy to screw up the GMPC, especially since the Game Master also controls the game world and plot. Thus, the GMPC turns into a plot device to keep the players on track, stopping them from going Off the Rails. Even worse, the GM can take too much of a liking to his PC, and the character slowly but surely turns into a Mary Sue; defeating all the enemies singlehandedly, doing all the cool stuff, and gradually reducing the players to supporting roles. He might also set up the adventure to suit his character, without considering the others.

This kind of GMPC quickly earns the ire of the players, since what's the point of even playing the game if all the DM's going to do is play with himself? They came to the gaming table to play as fantastic characters, crack some heads, and have fun; not sit and watch some Marty Stu kick werewolves through buses (And in the case of the Munchkin, he wants to be the one kicking the werewolves through buses). This may lead them to try to kill the offending character, which inevitably fails because of countermeasures including unstealable Rings of Invincibility and hostile allies getting struck by 10d100 lightning bolts. The Game Master may quickly find himself without a game if he doesn't get a clue.

An equally problematic variant is the GMPC who is obnoxious, absolutely useless or worse. These characters exist mainly to cause trouble for the players through their sheer incompetence, yet they are too pivotal to the plot and/or the fate of the world for the PCs to leave them to their justly deserved fate. This has all the problems of an Escort Mission in a video game, compounded by the fact that the Dungeon Master is doing it intentionally, rather than because of the inferior AI of a game.

It should be noted that any campaign setting that contains significant canon NPCs (such as the Forgotten Realms) may fall victim to this if the GM insists on having them travel with the party. Just as bad is the habit of using them as a Deus Ex Machina to bail out the party when they screw up, especially if you set up the PCs to fail on purpose.

Some games, on the other hand, require the Game Master to have a GMPC, usually a Non Action Guy with some kind of authority over the Player Characters.

There is a middle ground, a GMPC who behaves pretty much the same as most player characters, but you don't seem to hear much about them. Presumably because they don't make as exciting stories. A GMPC may be employed temporarily to stop new players from killing themselves before they've learned.

Not to be confused with Gnome Music Player Client.

Examples of games requiring GMPCs:
 * Bliss Stage's Authority Figure is one of these. Notable as one of the possible results of getting to One Hundred and Eight points of Bliss is for a Pilot to mutiny and take over La Résistance: If this happens, that pilot's player must become the new GM!
 * My Life With Master in a nutshell: The PC's in this game are the Igors to the Game Master's Dr. Frankenstein GMPC.
 * Maid the RPG is a sendup of the entire Meido trope in which the Master can have a powerful artifact known as the Desu Note. ...Do I really need to say it?
 * And the players' effectiveness depends on seeking out (or contending for) the Master's approval. On the other hand, the way the rules are set up, Masters tend to be pretty helpless in everyday situations (for the game's definition of "everyday"), and players usually have an explicit power to randomly derail plots they don't like, so it balances out.
 * It's notable that the author's self-insert GMPC gets accidentally killed by his maids. Twice.
 * The Wild Card in "Meddling Kids", a table-top game for kids based very loosely on mystery shows of the Scooby Doo kind.
 * Ninja Burger has the dispatcher, a Non Action Guy. His job is to watch the PCs on closed circuit camera and basically provide them with hints and assistance as needed.
 * Not a requirement, but Don't Rest Your Head has an optional rule where there is no assigned GM, and whoever gets the best roll in a scene gets to control the story, essentially making them the owner of a GMPC for a short time. You would expect this to cause problems, but considering the crowd that game draws in, it actually makes things quite interesting.
 * The Ring World RPG requires players whose characters mutate into protectors to give up control of the character to the GM. Protectors have superhuman strength and intelligence, but no free will, as they are ruthlessly devoted to ensuring the survival of their bloodline or species.
 * Paranoia not only has the Computer, but also specifically encourages the GM to be antagonistic towards the PCs and cheat when necessary.
 * Many Play By Post Game forums encourage the GM to have at least one character. Of course, these games don't always need a GM.
 * The Star Wars Saga Edition rules state that if a player accrues enough Dark Side points for his character to be fully committed to the Dark Side, he must hand his character over to the GM, who then uses the character as a villain NPC.
 * There are similar rules in the New World of Darkness concerning characters that hit 0 of whatever the game's Karma Meter is: Morality, Humanity, what have you. Generally speaking they are considered too impulse-driven to be capable of any meaningful interaction with other players.
 * As does Cyberpunk 2020. Lose all your humanity points and your character will snap. Since the easiest way to lose humanity points is to load up on inhuman cybernetic weapons, this is a very bad thing for anyone who happens to be in the area.
 * Also, in Don't Rest Your Head you turn into a monster if you use your Madness Talent too much.

RPG Webseries with GMPCs
 * Gandalf is a pretty hilarious example of the kind of things that go on with them in DM of the Rings.
 * Knights of the Dinner Table BA's are generally assistants (Gilead, Knobby Foot) ands put-upon Butt Monkeys whose sole purpose are to be kicked around by the PCs. The few times he had to resort to using an overpowered GMPC were done to hastily put the plot back on the rails.
 * The Noob has the MMORPG version, in which the Jerkass head developer plays a character. And cheats to try and win contests.
 * Darths and Droids uses this trope when the the regular GM can't make it to the session, and Pete takes over while still playing R2-D2, during the battledroid factory sequence. During the battle, he decides R2-D2 has rocket thrusters, which in the movie, pretty much allowed Artoo to save the day.
 * It otherwise averts the trope as the GM has had several NPCs travel with the party but none of them start behaving like PCs until a player needs a new character and takes over one of them (Sally does this a lot.)
 * Eluamous Nailo in Unforgotten Realms.
 * The Global Guardians PBEM Universe played with this. While there were loads of GMPCs, they were always characters being run in another GMs campaign.
 * While not quite a fit in this section, the MSF High Forum has a few of these. Mitchell, current head GM, is known for trying to make sure his stay out of the fight, especially at climaxes. For instance, Michelle was busy fighting her evil twin in an ultimately irrelevant battle (And lost), and includes one in his planned Mahou Galaxy adventure, and Legion-based online game.

Rare Film Example
 * Sir Osric in The Gamers: Dorkness Rising