I really like RPGs so this may be a bit RPG heavy...
Baldur's Gate I & II - More so the second one, greatest RPG for character development and interaction. So much dialogue done so very, very well.
Fallout 3 - Blew me away, story was great, action was great, my only complaint was I wanted MORE.
Oblivion - Fallout 3 improved on this one with companions and better plot, but Oblivion was and still is incredibly fun to play because of the amazing world around it.
Starcraft - I haven't played it in forever, but the campaign ending with the death of the Protoss leader was brilliant. A RTS with a great story? Seems almost unheard of, especially at the time.
X-Men Legends I, II, & UA - I consider this a trilogy of great games. This series proves how gameplay can over come the obsession with super-graphics. Also, truly epic storylines and great voice-acting solidify this series. I hope they just keep making them with different plots.
Alien Vs. Predator II - Favorite FPS of all time. Three different races, different play styles and stories. Exceptionally fun multiplayer. When are they going to make a current gen version of this??
Rock Band I & II - Party. 'nuff said.
Shining Force II - Favorite RPG from the Genesis and still a blast to play. Also want to see an updated version of this with the same battle system. Again, sometimes turn-based strategy done well trumps graphics and SF II was fantastic in terms of strategy.
Sonic II - Best Sonic game, even with Tails.
Final Fantasy VII & IX - VII has some of the greatest characters ever conceived (Red XIII and Vincent are still compelling to this day) and IX is a true fantasy with some equally cool characters (Freya comes to mind).
I don't rate any of these in comparison to one or another. I just came up with these off the top of my head. Honorable mention would include: Bioshock, Assassin's Creed, Halo Series, Prince of Persia: SoT...too many to think of...