Yes! You've come to the right place. No, this is not just another place with links to crumby shockwave/java games, but the percolated extract of the best stuff I've seen on the internet over "the years." These games require skill.

Best of all, these games are all free to play over the internet, but the game itself may cost some money.

Netrek
The classic CMU network game, the modern successor to xtrek. You fly around from an overhead view, controlling your small ship (of which there are different types ranging from scout to starbase). You shoot photon torpedoes and phasers; you carry armies on your ships to take over enemy planets. Your goal is to inflict the most damage on the opponent as possible.

The extensive and sometimes overwhelming control system may seem intimidating at first, but the game's steep learning curve also means that this game harbors a level of mastery achievable that few other games can offer. The best netrek players are undoubtably the best.

A Federation Starbase and a few escorts defend Indi.

Get started at the Netrek FAQ:
http://shell3.ba.best.com/~doosh/netrek/netrekFAQ.html

Subspace
A good, solid multiplayer asteroids deathmatch by Virgin, but that's the extent of it. Too bad it's commercial--well, actually, you can play for free, just with limited features. Still, not too bad a game, and certainly not nearly as steep a learning curve as netrek.

Still, the game involves less dogfighting and strategic skills IMHO. The graphics are certainly nice. Get it here.

Dink Dink.

Quake (& company)
If you haven't played Quake, then there must be something missing in your life. First person perspective deathmatch games have a wonderfully visceral and satisfying appeal to them, yum yum.

Better yet, Quake is so wonderfully popular and ingeniously designed that (a) you never have trouble finding large 30 player gib-fest on the internet and (b) there are so many modifications that games like Team Fortress are possible.

If you are lucky enough to own registered Quake on your computer, grab QuakeWorld now! Visit PlanetQuake. Visit ID software.

Modifications like Holy Wars make for
some pretty interesting deathmatches.

Check out Gamers Extreme to hone your Quake skills.

Starcraft
Starcraft is one of the current flagships of the real-time strategy genre of games. Heh, most of the people next to me in the cluster are playing it right now. If you haven't played RTS, then Starcraft is something definitely to try, bits of action with a larger, strategic game.

Terran bunkers fight off an organic alien race.
Positively delicious.

Check out Blizzard (makers of Starcraft and Warcraft), and BATTLE.NET, their free online Starcraft (and Diablo) service. There's a huge strategy guide at Gamers Extreme .