For me, Resistance: Fall of Man. If 2010 counts, then MAG. Resistance 2 and Killzone 2 come a close second.

Feels like i haven't logged in for a decade.

The 2000s had so many great FPS. Half Life 2 was good at it's time, though to me never among the greatest. Battlefield and CoD, though never my FPS of choice, both have their roots in the beginning of the decade. Monolith did awesome games like Aliens vs. Predator 2 and of course FEAR. STALKER though it had it's flaws was a unique and to me actually great game and we not only had Bioshock but also The Darkness. Not to mention Killzone 2 or Resistance.

But honestly, the 2000's FPS, that for me still play best, are the most enjoyable and replayable come from a series i actually didn't like at the beginning. Halo!
I just got into Halo in 2008, trying out the 360 backwards compatibility with that 'stupid' Halo 2, realizing it wasn't bad at all, though some years prior i really had disliked Combat Evolved. Today i couldn't even tell you why i didn't enjoy Combat Evolved in the first place. I can hardly decide if Halo 2 or Halo 3 is better though.

For my taste (I'm excluding here somewhat more serious tactical affairs like Operation Flashpoint, ARMA, Rainbow Six/Ghost Recon, Vietcong and such):

FarCry 1
Chronicles of Riddick: Escape from Butcher Bay
Return to Castle Wolfenstein
No One Lives Forever 2
Star Trek Voyager - Elite Forces

As a fan of HL1, HL2 was fine, but didn't made me as awed as the first one.

Halo: CE, by a mile. It is the game that really moved the FPS genre to the forefront of gaming. It's also just a generally excellent game. I just replayed it last week, and it still holds up well.

I think Halo: CE is the most influential FPS of all time.  I can't even think of another game to put in the conversation.  

Would you kindly vote for Bioshock as it's the right answer.

Biggest ones that come to mind that I played the absolute crap out of were Halo 1-3, Team Fortress 2, CoD 4: Modern Warfare, Bioshock, and Counter-Strike: Source

That's hard to answer, the 2000s had so many great FPS.

Halo(2), CoD4, F.E.A.R, Team Fortress 2, Portal, Far Cry, Crysis, UT2004 were all fantastic games but I will choose Half Life 2.

Yeah HL2 is the safest pick but It was an incredible experience back then.

I forgot crysis damm

Halflife 2
Bioshock
Crysis
Borderlands

And not sure if it counts, but Metroid Prime.