CounterStrike! I think that's the best FPS game of all time because it's the game that got me into first person shooters. The game was simple - it just pit terrorists against counter terrorists. Yet the gameplay was so good that it kept me and my friends glued to the monitor for hours on end.
On mobile, my favorite shooter is ShadowGun DeadZone. It's such a balanced third person multiplayer online shooter. It's easy to control even on the small screen, has great graphics and animation and awesome gameplay. It's also free so that's another plus for the game. If you have a decent mobile device, download this game.