For Story/Single Player
Halo CE, I sank DAYS into that game when it came out.
Half-life, sadly I only managed to play that brilliant game last year. It was a pleasant surprise mixing the challenge of an ordinary FPS with puzzles, horror/thrills and SCIENCE!
For multiplayer
Call of Duty: World at War. Myself and 2 of my friends where an unstopable force on that game, last true CoD game in my opinion. Teamwork really counted. Now in CoD, you can do just as well on your own with all the perks and stuff they introduced making each soldier a one man army.
And Battlefield: badcompany 2
I didnt have as much fun on this since I only got to play this with friend rarely. But makes up in the teamwork department what recent CoD games have lacked.
Overall Best FPS
Timesplitters Future Perfect. I still play that alot, has stood the test of time quite brilliantly.
Single player, challenging, balanced, funny. It lacks nothing except a jump button. The singleplayer challenges are great little distractions once you finish the story.
Multiplayer, tons of characters to play as. Fully customisable weapon sets and bots. Since Original Xbox live has died and many great multiplayer games along with it, the fact that it has AI bots that are not completely stupid and are actually fun to play against makes up for that. The tournement mode offeres a real challenge when you try to get Gold medels on everything!
And finally the mapmaker, While fairly simple at heart, the ability to make indepth story missions and wild multiplayer maps allows you to add so much content to the experence!