Any FPS by Valve.
Also, ones I haven't seen crop up too often in this thread, Goldeneye, Perfect Dark and Timesplitters, especially Timesplitters.
I'm quite surprised it's had little mention. Personally, I think it takes some of the best bits of Goldeneye and Perfect Dark (which, to the best of my knowledge a group of developers who worked on Timesplitters also worked on both those games), the campaign mode of the second and third were top notch in terms of gameplay (although I admit they suffered from having very little in the second to only some almost random plot in the third), the arcade mode was incredibly fun and it was good to see bots, however stupid they were, to be in a game's arcade mode, and it had a pretty decent mapmaker in all three games which gradually got better and more developed. I still occasionally take out my PS2 to play on them, especially the second because even to this day I've been unable to complete the campaign missions on hard (Managed to finally do Siberia on hard after much trial and error though, and I've heard by many people's opinions this was one of the hardest levels in the game, if not THE hardest).
My next favourite will have to be Call of Duty 4, simply because of a well thought out control scheme on both console and PC (Although surprisingly I found the preset console controls much easier than the preset PC controls), and it's blend of gameplay and storytelling is surperb and that technique should be used in a lot of other games. I hope Modern Warfare 2 doesn't mess up everything the studio learnt in its transition from the previous CoD games to the fourth installment. I feel so ashamed that I sold the copy I had for my 360 to someone a while back. I wish I could play it again now, especially online.
Halo is also up their with my favourite games. I habour the bitter hatred towards the second game that even a lot of fans I know have, and personally the third one was mostly an unenjoyable experience, both in single and multiplayer. However, the first game I found strangely addictive. I couldn't care less for the story in the game, since I was too busy shooting and driving to care what Cortana had to say, and the way the story was given to the player made it seem like more of an afterthought than an original idea for the game. However, I found the combat sequences, if a little slow because of Master Chief's criminally slow snail pace of a walking speed, genuinely enjoyable at the time and still love going on the game randomly to relive some of those experiences. Even better though is the multiplayer on the PC version. A good 8v8 on Blood Gulch with Warthogs, Scorpions, Ghosts, rockets, grenades and bullets flying everywhere is always fun.
Like I assume most people had done the same, I greatly enjoyed Painkiller at Yahtzee's recommendation. Great going back to Doom-styled gameplay, same reason I love games series like Unreal Tournament and Quake with their similar styles. Only one thing I want, and that's a mod which adds tits and fire to the lightning and shuriken gun.
I quite like the title of this thread since it uses FPS as a plural, so I don't feel like I'm off topic by stating why I like many different FPS games. I doubt there will ever be one true, definitive FPS game that fits everyone's enjoyment as the whole idea of the FPS genre is down to opinion, either you like that style of game, or you don't. Although I'm certain a combonation between Unreal Tournament, Half Life and Timesplitters would be very close to that.
Other FPS games that I simply don't have time to describe why I like them so much more over others:
STALKER series
Postal 2
Tribes (Specifically Tribes:Vengeance)
There are possibly others I've missed, but oh well.