Vanguard1219 said:
My recommendations are American McGee's Alice and the Longest Journey, provided that you can find a copy of either of them.
The latter, at least, is available on Steam.
You can find a great deal of free games that don't have especially high requirements on independent games websites such as IndieGames [http://www.indiegames.com/blog/] and the Adventure Game Studio database [http://www.bigbluecup.com/games.php]. Definitely check out these sorts of sites.
For pay games, GOG [http://www.gog.com/en/frontpage/] is going to be your best resource. Check the system requirements, though, because some of these games are relatively recent.
You said that Thief 1 isn't working. This is probably because you're trying to run it on a dual-core computer, or maybe it's just not agreeing with your XP or Vista install. Check this [http://www.ttlg.com/forums/showthread.php?t=75031] FAQ, and you will probably find a solution to your problem.
As for specific games, everybody's listed all of the ones I would suggest, except for these:
Half-Life - still one of the greatest shooters ever made. Also Opposing Force and Blue Shift, plus the mountains of fantastic-quality Half-Life mods - single- or multiplayer.
Giants: Citizen Kabuto - a seriously underrated action game that's really difficult to describe. It's available from Gog.com.
System Shock 2 - a great, creepy sci-fi action/horror/RPG thing. Bioshock is just a heavily simplified and thematically transplanted version of System Shock 2.
I'm also going to get around to the Baldur's Gate series and Planescape: Torment sometime before I die, dammit. From what I hear, you should, too. (There is a mod out there that lets you play Baldur's Gate 1 in the Baldur's Gate 2 engine, which means that you get to play with all of the improvements of BG2 in the first. You need both games to do this, though.)