Going to list some of the indie games I have (and I'll probably miss A LOT of them)

3079
A Valley Without Wind
A.R.E.S
AaaaaAAaaaAAAaaAAAAaAAAAA!!! For The Awesome
AI War: Fleet Command
Alien Shooter
Alien Shooter 2: Reloaded
Amnesia: The Dark Descent
And Yet It Moves
Aquaria
Atom Zombie Smasher
Audiosurf
The Ball
Ballistic
Bashi Blocks
Beat Hazard
BEEP
Ben There, Dan That!
The Binding of Isaac
BIT.TRIP.RUNNER
Braid
Breath Of Death VII / Cthulhu Saves The World
Bunch Of Heroes
Cave Story+
Costume Quest
Crayon Physics Deluxe
Critical Mass
DeathSpank
Defense Grid: The Awakening
Dungeon Defenders
Darwinia
DEFCON
Dungeons of Dredmor
E.Y.E Divine Cybermancy
Fate of the World
Fireball
Inferno+
Koya Rift
Galcon Fusion
Gemini Rue
Gish
Gratuitous Space Battles
Hacker Evolution
Hacker Evolution - Untold
Hacker Evolution Duality
Hammerfight
HOARD
inMomentum
Jamestown
Machinarium
Minecraft
Nation Red
Obulis
Osmos
Pineapple Smash Crew
PixelJunk Eden
The Polynomial
Runespell: Overture
Saria
Samorost 2
Sanctum
Scoregasm
Slydris
Soulcaster
Soulcaster 2
Space Pirates and Zombies
Starfarer
Super Crossfire
Super Meat Boy
Terraria
Tidalis
Time Gentlemen, Please!
Trine
Towns
Uplink
VVVVVV
World of Goo
Xotic

That'll do... as I said... there are probably A LOT that I own that I've missed...

Octodad. It's free and all you need is a mouse to control it.

http://www.youtube.com/watch?v=lVoSYDWX2Ig

To the moon was pretty good.
http://www.youtube.com/watch?v=sqkJuSV-23U

Tried to read through all of the suggestions and will post some that I didn't see:

Bastion - Not sure that this counts as an "Indie" game but it has that feel (clever, smaller levels, hand crafted environments). It's not strategy but the action-RPG is still enjoyable.

Serious Sam: The Random Encounter - This was actually pretty entertaining to me.

Stacking - Just released for PC. Again, not as low-fi as many indie games but still very enjoyable.

SpaceChem is extremely entertaining if you have the mind of a programmer -- it's basically building an assembly line with a hard-coded program for creation of particular molecules. Comes with an editor and ability to download community-designed puzzles.

Sanctum is fun if you can play it with a friend or two (or three) -- if one of you prefers, say, the close combat FPS aspect, one sniping, and one the "tower defense" placement, you'll have a great time.

If you're more into RPG-like character improvements, then I'd definitely recommend Dungeon Defenders (not sure if this counts as indie), again to play with your friends. I'm a grown-up man, and this game sees me stupidly smiling and giggling for the duration of the whole match. Extremely relaxing and invigorating. :-)

I would like to recommend Cthulhu Saves The World and Terraria! :)

1. Rock of Ages
2. RUSH

I'm making a lot of Posts here, I know...but having Steam has introduced me to a whole World of Indie games I never knew were out there...

I must second the suggestions for Cthulhu Save the World (hilarious and purposefully made to look and sound like an Old School 16 bit console game) and Avadon: The Black Fortress.

Avadon is surprisingly big and has a Gameworld with a lot of really good background. IIRC it is just the 1st part of a larger series they've planned.

Samorost 1, 2
Machinarium
The Old Tree
Braid (<- amazing soundtrack)
CStW & BoDVII
Audiosurf

I would say defense grid is probably the most polished/best "standard" defense game.

More creative defense games are like revenge of titan or sanctum.

don't enter this post into the draw my 2 recommendations are in another post sorry.

I can recommend Minecraft.

Recommendation:

Braid, you have to have to try this game at least, its the only game truly impress me in indie, although spacechem is great puzzle game Braids way of puzzle is just creative/different and feels very rewarding.

Osmos, maybe only good for an afternoon but once again first time trying it was like wow cool.

One game I would recommend for people who like rouge likes would be Desktop Dungeons. It is currently in its beta phase and is constantly updated but is a very fun game. You just start a map as a race/class of your choosing and try to kill the boss or die trying. The maps can take about 10 to 15 to either win or fail.

If this game interests you the website is http://www.desktopdungeons.net/ I would also recommend trying out the free alpha version to see if you would like it.

Edit: Also if you get Dungeons of Dredmor you will not regret it. The game is incredibly fun, but I would recommend getting the dlc as long with it since is provides extra skills you can pick for your character.

Another quick recommendation would be Recettear: An Item Shop's Tale. This is a kind of weird dungeon based RPG where the main goal is to collect items to sell at your store to make money to pay off a family debt. The game is really fun and simple to learn and get into.

As a RPG fan, I recommand Spiderweb's RPGs. Avadon and Geneforge series are on steam, while Avernum 5 and 6 are on GG. The better way is to first go to spiderweb website, download some demos and try. The demos are actually full game, but with later areas locked. It is a bit more expensive there than on Steam, but the older titles are only available on their website. Once you buy it, you'll receive a code to unlock the demo and gain access to all areas so you can have the full game and continue on, meanwhile they'll also mail you a physical copy.

These games are short and very good:
(free non-HD version [url=http://zarat.us/tra/offline-games/eversion.html]here)
CreaVures