Team Fortress 2 is a pretty challenging FPS for new comers, and it's Free-to-Play. Unfortunately, there's not much story to it.

Dead Space and Dead Space 2.

Fallout and Fallout 2

Mass Eff- damn, any RPG from BioWare, except Dragon Age II.

Most people think Baldur's Gate II is the best. They both have positive points. I'd recommend playing BG1 first, as it's basically one long story, going from BG1 to Shadows of Amn and ending with Throne of Bhaal.

You should also install http://www.pocketplane.net/mambo/index.php?option=content&task=blogcategory&id=143&Itemid=98, which will allow you to play BG1 in BG2's engine, as the latter implemented some important improvements.

Metal Gear Solid 4: Guns of the Patriots
Borderlands (play it with a friend!)
Red Faction: Armageddon
Fallout
Super Mario Galaxy (though I hear 2 is more difficult)
Batman: Arkham City
RAGE
3D Dot Game Heroes (Zelda with a few interesting twists)
Saints Row the Third

And that's all I can think of offhand.

Minecraft.

I really enjoyed the first Dragon Age.

skyrim
super meat boy (very hard platformer)
titan quest + immortal throne (similar to diablo)
need for speed - shift
need for speed hot pursuit (2010)
kingdoms of amalur (try the demo) releases early feb
baldurs gate 1 & 2 (old but very good rpg)
crysis 2
bioshock 1 & 2
saints row the third (similar to GTA)
deus ex (the original)
team fortress 2 (free)

You should play red dead redemption.

I was actually surprised some people didn't like the game because "the whole thing takes place in a tunnel."

I'm just like...you missed the whole point...

Final Fantasy VII

Transport Tycoon deluxe

Skyrim.
STALKER.
Dragon Age: Origins.
If you like Kotor, then maybe the old republic, but it costs money so you know...
Far Cry 2.
Mafia.
Mount and Blade, any game will do, but I recommend Warband.

Alan Wake

If you'd like a stylish, sleek, fast, and really really really good looking third person shooter, I'd recommend Vanquish to just about anyone! It's just awesome, and an unfortunately overlooked gem.

Not one person said
The witcher?
For shame.

Although to be honest, AoC and STO have impressed me so far with both the visuals and the story.
Man when I got my XFX 7970 in my pc and I modded skyrim up the graphics where amazing, so was AoC and STO.

The withcher 1 and 2 are pretty good.
Drakensang.

http://en.wikipedia.org/wiki/Drakensang:_The_Dark_Eye

http://www.thewitcher.com/

http://en.wikipedia.org/wiki/Vampire:_The_Masquerade_%E2%80%93_Bloodlines

Dragon age orginons was good IMO, along with Ice wind dale 1 and 2.
Baulders gate 1 and 2.
Mass effect is pretty neat.
X2 the threat and X3 as amazing for a sandbox game with a good story.

Also, Earthbound.

Skyrim? I'd also suggest the original Fallout's and the older TES games.

Halo 3?

http://store.steampowered.com/app/39510/?snr=1_7_7_151_150_1 seems to fit the description. Except "fresh" and "new" maybe, but thats relative.

Its similar to TES games, but more linear and a lot more challenging. It has great atmosphere and characters, a decent storyline and should give you plenty of hours of entertainment, if you dont mind the dated graphics.

vampire the masquerade bloodlines, pretty good rpg.

Gothic 2, seconded.

Pathologic. I look forward to the ranting afterwards.

2nding the Witcher.
2nding Metro 2033
Adding a recommendation for Saints Row The Third. This game is nothing but ridiculous fun.
Also if you haven't yet look at getting Golden Eye: Source. I've been playing it like mad since I found it. Its amazing to be back in bond.

Although I cannot bring myself to like the first one, The Witcher 2 reminded me of how I wasn't at all hyped about Fable and picked up the Lost Chapters on a whim and got the 3rd best RPG experience on the original X-Box. Also, playing the witcher one is not at all necessary to understanding the second. I find the ending annoyingly inconclusive however (in the dragon age II sort of way)

I rented metro 2033 and found one of the most detailed games i've ever played (for instance you do not "loot" corpses, you pull the items off of their vests.)

Saints Row... Well, my character is the joker who shoots fish guts at people then a sewer shark erupts from the concrete and tears them in two.

Golden Eye: Source... ITS GOLDEN EYE ON THE SOURCE ENGINE!!!