Single Player:
Half Life: Source
Half Life 2
Fallout 3
Fallout New Vegas
Multiplayer:
Team Fortress 2
Left 4 Dead / Left 4 Dead 2
I'm sure you will get a chorus of BF3, MW2, Counterstrike: Source, etc. as well but I'm just listing the FPS's I most enjoyed since Quake 1.
If you like great looking games try Syndicate, Dead Space 2, Crysis 2, Resident Evil 5, Mass Effect 2/3 .... and BF3 for online play.
Mass Effect is technically not a first person shooter but I liked ME1 and haven't played ME2 or 3 yet.
Also, Syndicate? You mean the 1993 version or something else? The latest Syndicate is nothing like the original and has poor reviews.
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Crysis 1 (not available on steam)
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Crysis 2
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