What is a great pc online game(see details)?

Splinter Cell conviction has a fun CoOp mode. But you will need Tunngle to find any matches.

You could try Alliance of Valiant Arms.

A game that seems right up your alley is Team Fortress 2.
It is a class based First Person Shooter. You pick from 9 different classes to play, and easily switch between them. Best of all, you can play for free.

Team Fortress 2, Battlefield: Bad Company 2 (my personal favorite), and Left 4 Dead 2 get my vote for shooters, and Starcraft II and Warhammer 40K: Dawn of War II get my vote for strategy.

StarCraft2.

Supreme Commander is a pretty good RTS to play with or against other people. Portal 2 co-op is well worth a look.

World of Tanks is a decent one, and it’s free. Yes, you can buy stuff, but the game is perfectly playable even if you don’t spend a dime.

Crossfire is a decent FPS… unfortunately there’s a lot of hackers… :( but it’s free, and has a fairly wide selection of weapons.

Starcraft II seems to have quite the following

@Ltryptophan The big problem with Starcraft and Starcraft II is that you often face people like this.