FPS- "First Person Shooter" A game where the players walk around shooting eachother or a computer enemy. View is from the eyes of the character.
RPG- "Role Playing Game" A game in which players assume the role of the character, and play as that character would. View is often 3rd person, but could be 1st. Areas are usually large and there are many directions that a character's abilities can be taken.
Strategy- Games in which strategical decisions have a significant role in the outcome of the game.
Other- There are many games that are very unique and do not fit into any common type.