Seeing as how I've been PC-gaming for most of my life, I can give you a heck of a lot of suggestions.
First of all, look into Steam. http://store.steampowered.com/
I can guarantee that you'll find a lot of things you'll like there.
Then, there's also GoG. http://www.gog.com/
They hold mostly "Good old Games" made compatible with modern systems, but, they've recently gone for more recent stuff as well.
Now, then... onto some actual names, here...
Just about anything from Valve, such as Team Fortress 2, like the first guy mentioned, with much gusto. It's a multiplayer first-person shooter... with class. It came out in 2007, yet it still receives updates and plays like crazy to this day. And it's just so happy and pristine!
Half-Life is a more varied, slower-paced first-person shooter. And, boy is in ground-breaking! It tells a fantastic story of Dr. Gordon Freeman's escape from the Black Mesa laboratory after an experiment goes horribly wrong, and the complications that arise over time... all without the use of cutscenes. It actually works out really well.
Portal. This is a smaller game. But it is intelligently-designed and such a joy to play. I'd recommend you get it in "The Orange Box", which is a collection of Valve games. It also includes Team Fortress 2 (mentioned above) as well as Half-Life 2, along with it's "episodes". It's not actually a shooter. Well, not with bullets, anyway. It's a brain-bending, dimension-breaking 3-D puzzle game that took the world by storm. For some reason it's grouped with those other shooters in The Orange Box, and yet... it somehow doesn't seem out of place.
Zapper: One Wicked Cricket! It plays similarly to some older games. Namely, Chip's Challenge with a hint of Spyro the Dragon and a bit of Frogger off to the side. It's a good action-adventure-puzzle-platformer that should keep you busy for a while.
007. In my opinion, the best titles were the ones developed by Electronic Arts. And there are certainly more of them, such as NightFire, Everything or Nothing and Agent Under Fire, as well as some others based on the films.
Plants vs. Zombies. It's a violent, tile-based, real-time strategy tower-defense game. And it's quite original and gobs of fun. Hope you like slaying zombies!
Left 4 Dead. Speaking of zombies, there's another game from Valve that I neglected to mention. This one! It's an adrenaline-pumping teamwork-based zombie slaughterfest. There are two games so far. I'd recommend you buy the second one, though. Why? Because, there are add-ons for the second one that let you play every single aspect of the first, with melee weapons added and the three (or four, counting the mudders) new zombies from the second game.
[Prototype]. This is a sandbox game that is somewhat like Grand Theft Auto meets Assassins Creed... with super powers.
I'll leave you with that.
BioShock. This choice-filled, underwater guns-and-mutations shooter may keep you on your toes. Choose what you do with care.
Alan Wake. This is a rather mysterious game about a writer searching for his missing wife. Trust me, it's a lot better than that sounds.
Pretty much anything by Id Software! Seriously, these guys have made some of the most legendary first-person shooters in history. As a matter of fact, they invented first-person shooters! The very first one ever made was Wolfenstein 3-D (a new one came out the other year), but it was really Doom that everyone remembered the most. It was revolutionary! Though, personally, I think their best work was the Quake series. Although, I still have no idea why the first game takes place in a medieval castle, while the second one is up in space.
SERIOUS SAM. Because, this: http://www.youtube.com/watch?v=JTbwOukQlDY