RTS - Real Time Strategy. Games that involve the creation of units needed to overtake an opponent while avoiding the same fate.
Examples: Command and Conquer, Starcraft, Age of Empires.

RPG - Role-Playing Game. A game that puts you in the shoes of an already-created character. Most RPGs revolve around the character trying to stop a plot that could lead the world of the game into oblivion.
Examples: Final Fantasy, Phantasy Star, Legend of Mana.

FPS - First-Person Shooter. These games put you in the shoes of the main character...almost literally. In a FPS, you see through the eyes of the character, only seeing what they look like through cutscenes or mirrors. Though pretty much speaking for itself, the character of these games is usually given a payload of weaponry to dispatch his/her enemies with.
Examples: Duke Nukem 3D, Doom, Halo.

Sim - Simulation. These games allow you to control the game itself...from creating cities to destroying lives, a Sim gives the player the power of God.
Examples: The Sims, SimCity, Airport Tycoon.

MMORPG - Massively Multiplayer Online Role-Playing Game. Take the base elements of a standard RPG, throw away the Plot concept along with the Main character and the Ending, and you have an MMORPG. MMORPGs tend to be overpopulated, slow, and after a while, they get to be boring.
Examples: World of Warcraft, Ragnarok Online, Ryzom Ring.