I refused to reply to this thread because there's too many answers.
And FPS is not a good term.
Singleplayer-experience-oriented? (Duke 3D, Doom)
Multiplayer-experience-oriented? (Battlefield 1942, Tribes 2)
MMO? (Planetside, WWII Online, a myriad of titles coming in 2009/2010...)
Arena FPS? (UT, Q3A)
FPS Hybrid/Crossover? (Metroid Prime, Mirror's Edge)
FPS/RPG? (Deus Ex, System Shock 2)
Tactical? (Rainbow 6 before it was dumbed down)
There's more subgenres that I can't just think of here.
There's way too much in the genre, and too many beautiful examples of all, to really just say "best fps." And there's plenty of great single player games that work perfect for multiplayer (Goldeneye)
Which reminds me, on N64 alone, I can think of an arena FPS (Turok: Rage Wars), and a single-player/story based FPS like Perfect Dark....
It really comes down to personal preference, especially in storyline, setting, characters, and weapon selection, for each category.
Or if you want the ever-encompassing FPS, you might as well just play CoD 4 since people call that realistic and find the multiplayer fun and interesting.