My favorite first person shooter has definitely got to be 10SIX (Now known as Project Visitor) and GUNZ. Both had amazing twitch requirements and 10SIX stood out from all others because of its amazing concept.
My choice of Best Arcade shooters would have to include Battlezone (1980) and Star Wars (1983) (though Star Trek - Strategic Operations Simulator (1982) is a close third). I absolutely loved those vector graphics games and pumped quarter after quarter into them. I can still remember the digital sound of Luke Skywalker saying "This is Red Five standing by".
For the PC, I think my top choice is Quake (1996) on a Diamond Monster 3D add-in video card with the VooDoo chipset. I remember being blown away the first time I played a game with Hardware rendering. Rest in peace, 3DFX.
Second top shooter for me is Half-Life (1998). Enough good things cannot be said about how high Half-Life set the bar with its immersive story telling and crowbar wielding action. Those damned Headcrabs always gave me "itchy-pits".
F.E.A.R. - Rocked, second not even close. I still play the multiplayer nearly every day.
STALKER Clear Sky - awesome, can upgrade weapons puts it over the first one. More open than COP.