Ultima VII and Ultima Underworld:
WAY ahead of their time. Some of the best RPGs ever. Give you that "living, breathing world" feeling. Play with DosBox.
X-Com, Terror from the Deep:
The creepy pinnacle of turn based squad strategy. I prefer "Terror from the Deep" to "X-Com UFO Defense", because TFTD is creepier and has a more threatening, serious tone to it, whereas the UFO Defense atmosphere is too cheesy for my taste. Play with DosBox.
Star Control 2:
Unique space adventure with real time ship combat. Amazing game, great story. Play the remake for modern operating systems: "SC2: The Ur Quan Masters".
Alpha Centauri:
One of the most polished and insightful civ-games ever. Like most early civ games it suffers from poor AI, but the background research that went into this game and the setting are awesome. Usually runs on Windows without problems.
Indiana Jones and the fate of Atlantis:
IMHO one of the best Lucas Arts adventure games. Play with DosBox.
X-Wing and TIE Fighter:
The graphics have aged, especially with X-Wing, but the thrilling gameplay of two of the best space combat sims ever is still there. Play with DosBox.
Doom:
I would almost consider it a sin for any gamer not to have played the original Doom. Play with DosBox for the original feel, or use one of the numerous remakes for better graphics & usability.
Colonization and Master of Magic:
To me, the outdated graphics add to the charm of these fun and thoughtful civ-spinoffs. Run with DosBox.
(the original) Street Fighter II:
Still one of the greatest beat-em-ups. It just "feels" right. Play with a SNES emulator.
These are some of my favorite games of the past, that I still fire up every once in a while.