Deus Ex. THE gold standard of gaming. You are obligated to play this.
Diablo 2. Do I even need to explain?
Neverwinter Nights 1, plus both expansions.
Morrowind, plus both expansions. Forget the dumbed-down shiny-graphics Oblivion, Morrowind is what the Elder Scrolls series is all about--complexity and depth.
Freelancer. Great gameplay, great story, a beautiful galaxy to explore, and a huge and comprehensive (even still!) modding community. How can you go wrong?
Total Annihilation (Looking for an old-school RTS that makes Starcraft look like crap by comparison? TA is it--and it's even older than Starcraft is)
Giants: Citizen Kabuto (A very entertaining and under-appreciated third person shooter/action game with some great humor)
Jedi Knight 2: Jedi Outcast--and/or Jedi Academy. (The Force Unleashed was a turd, well-polished with fancy physics and effects. Jedi Outcast is what a Star Wars game should be)
Hitman: Contracts. Not as good as Blood Money, but it'll run smooth on your system. Fun, replayable, but probably the lowest priority game on this list.
Out of those games, Deus Ex is the most important. Every gamer has to play it--period. There is no excuse for not having played Deus Ex. Morrowind and Jedi Outcast are also games that absolutely must be played.