Wow, this is hard, I've been playing for so long. When I was a kid I had Commodore 64 and before that some crappy console, and I've been playing ever since.
Of course games with a story and atmosphere are great. My all-time favourites would be Dune (the first one), Fallout 1&2, System Shock 2, Doom (not much of a story but....you know, it's Doom), Gabriel Knight 3, The Dig, Blade Runner, Mass Effect, The Darkness...there are just so many.
Some light strategy games also had their impact on me. Civilization is a given, and also Close Combat games, I had them all except the first one.
Championship Manager 2 was the first proper manager game I had, although I never cared for football, this game was just great. As a hockey fan, NHL EHM 2007 remains the best hockey manager game ever made.
Then, sports games in general. I really don't care about anything else than hockey, but man I've had fun playing PES 5&6 and FIFA 09 with my friends. NHL 10 stands as the best hockey-in-action game. Also, the eternal Wayne Gretzky Hockey must be recognized.
Racing. I'm not the guy who gets excited about cars, but I've had a buttload of racing games. Forza Motorsport 2 and Project Gotham Racing 3 were the most recent hits for me, earlier would be Stunt Car Racing (back in C64 days).
Then, shooters and action games. There were some awesome 2D shooters back then. Turrican 2, Gods, Switchblade 2, R-Type, X-out, Denaris, Armalyte (C64), Prince of Persia, ...too many. 3D shooters nowadays...a lot of good stuff, but can't get so excited about them anymore. The Darkness was great, and R6 Vegas games have remained ones of those I still play on Xbox live with my friends. GTA4 and Saints Row (the first one) rule the sand box games.
Let's not forget about the good old beat 'em ups. International Karate with it's sequels, Panza Kick Boxing, Bruce Lee! you name it.
I am old.