I would recommend Atom Zombie Smasher to just about any gamer with a PC or Mac.
Bastion, to see how much content can be packed into a single downloadable
Braid, to provoke reflection while playing a not at all bad platformer
Super Meat Boy, to push your skills to the limits
Puzzle Agent, to see how a seemingly innocent puzzle game can sustain an uneasy and creepy atmosphere while uncovering one of the best intrigue based stories to be found in gaming
I am not the type of person who thinks that all games are for everybody, and i dont think every one should play a game just because i like it. but i have been playing Cities in Motion, and i think if you like Railroad tycoon, or especially Transport tycoon you should try this.
There have been a ton of great titles mentioned, so I'll try to keep mine a little different. I don't know exactly if these would qualify as indie - some of them had pretty big releases for their platform's online stores, but they're defnitely worth checking out.
- Joe Danger - Fantastic time sink of a game, with a lot to do in very manageable chunks. Super fun and some spot-on controls, too.
- Shadow Complex - I'm sure everyone here has at least heard of it, but for Metroid-esque games this generation, I think it can't be beat.
- Torchlight - Right now, other than Titan Quest, it's the best Diablo game not named Diablo.
- World of Goo - Stupidly fun, great amount of charm, and gets surprisingly clever at times.
- Hector Ep. 1-3 - I know these were published by Telltale, which I don't consider to be very independent, but they were developed by a small team and are definitely indie-spirited. Very fun point-and-click adventures.
- Space Rangers 2 - Again, not really truly indie, but has a fantastic amount of indie spirit. It's really a bizarre mix of several different game types with, ostensibly, an arcadey-space adventure type feel. If you haven't checked it out, it's crazy fun.