*adjusts monocle* Strongly recommended ones are in bold.
Operation Flashpoint: Cold War Crisis (2001)
Arcanum: Of Steamworks and Magick Obscura (2001)
Max Payne (2001)
Clive Barker's Undying
The Elder Scrolls III: Morrowind (2002)
Beyond Good & Evil
Tom Clancy's Splinter Cell (2003)
Max Payne 2: The Fall of Max Payne (2003)
Tom Clancy's Splinter Cell: Pandora Tomorrow (2004)
Rome: Total War
Tom Clancy's Splinter Cell: Chaos Theory (2005)
Sid Meier's Civilization IV
Medieval 2: Total War (2006)
Also, while Portal (1) came out in 2007, it should easily run on your hardware. Half-Life 2 is a good game, and well-recommended. It has two episodic sequels, which should also run well.
Are games from 2000 okay? If so:
American McGee's Alice
The Longest Journey
Prince of Persia: The Sands of Time
Supposedly Yahtzee's favourite game, or something. I was never a fan of PoP and the general old-school platforming genre in general, so I didn't like too much.
Star Wars Jedi Knight II: Jedi Outcast
Star Wars Jedi Knight: Jedi Academy
A ton of fun, nothing world-changing, but just a lot of fun to be had here.
Haven't played it, but it's considered one of the greatest RTSs made and got some extremely high scores. I should get around to playing it sometime, as the sequel has just been announced.