'puts on old man suit'
All these young'uns and their new fang-dangled shooters with their billions of polygons and specular lighting shaders. A few sprites and an animated hand holding your gun, that's all you need!
One can't simply point to one single "best FPS ever". There's never truly been one perfect game, but there have been quite a few FPS's that have paved the way for all others since.
Doom - though not the first FPS, it essentially cemented the genre in gaming.
Quake - set the model by which most multiplayer or online shooters adhere to today. it's list of game modes from deathmatch, team deathmatch, ctf, etc, are industry staples.
Half-Life - set the standard of narrative by which all decent FPS's are made. it was a near perfect example of how well gameplay and story could be blended together seamlessly and continues that tradition today
Goldeneye 007 - basically defined what and how a console FPS should be made. tight controls, insanely addictive split-screen multiplayer and a passable story mode.
System Shock 2 & Deus Ex - prime examples of how best to blend FPS gameplay with RPG like story telling and exploration. without games like these, there'd be no Bioshocks
Portal, Metroid Prime, Mirror's Edge, and a few too many to list - there's been only a few in this last list. some critically successful, some not, but the one thing they all share is that they showed that the first person shooter genre isn't limited to just running and gunning through corridors or just about killing nazis, cyborgs, or aliens. They showed that the FPS genre can be original, creative, endearing, and above all, ground-breaking.
Lastly, I'm pretty sure this thread's been done to death. Please try refraining from inundating the forum with "what's the best ****" or "which is better?" threads.