FPS stands for first-person shooter, a video game genre focused on modern or futuristic action and, well, shooting.
FPS games are designed using a first-person perspective, with the screen showing the game environment as it would appear if you were the character. They also usually show what the character is holding in their hand(s), even though people don't normally hold their hands at approximately shoulder-level when walking around. They often include a heads-up display including a health meter of some sort, something to indicate how much ammo you have left, and other vital statistics.
FPSes tend towards having little if any plot besides "there are bad guys, shoot them"; just enough to get the story moving. Some, however, are very deep, with surprising twists and turns revealed as you explore the world.