The first person shooter (FPS) is a actualy a sub-genre of shooter games. The term started to be used widely in the 1990’s (circa 1992-1993 when Wolfenstein & Doom was released) and is generaly used to describe any game with a first-person view.
The first-peron view tends to be with the characters hand/gun shown in the center (or just off center) of the screen.
The first person shooter was really established with the release of Wolfenstein & Doom, most early first person shooters are compared to Doom and many decended from the Doom engine.