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.
Call of Duty Modern Warfare 2. Its very obscure, you posers won't have heard of it.
Or Binding of Issac. Oh and my GOTY Bastion.
Dirty hipster! I knew about Modern Warfare when it wasn't half as Modern as it is nowadays..