There's an online game called Super Smash Flash 2 that you can play over the internet. No download required, just play it right off the website. It's basically a pixelated version of Smash Bros. That also includes a number of characters that don't appear in Smash Bros. Series like Sora (Kingdom Hearts), Ichigo (Bleach), Naruto (Naruto~) and Goku (Dragon Ball)
It's pretty simple and fun. You can play it two-player on just one keyboard as well.
http://kbhgames.com/super-smash-flash-2/
There's also a game called Battle For Wesnoth, which is a sprite-based strategy game set in a fantasy world. Pretty simple, has several long campaigns with storylines and characters, a fun map editor and random skirmishes, online multiplayer, etc.
It's free as well, and you can download it here:
http://www.wesnoth.org/
Another game I like to play is Dawn of War and it's sequels, Winter Assault and Dark Crusade. They don't seem to take much to run, and you can find CD keys easily if you buy a second-hand copy. It's a disc game, but you might be able to download it if you have Steam. The Steam version will allow for mods and multiplayer as well, I think. But for just the game itself, if you can find a disc it will run. It's a top-down military strategy in a similar vein to Age of Empires or Battle for Middle Earth, but with smaller bases and larger battles. It's set in the Warhammer 40,000 Universe, so you get to play as Space Marine factions, Orks, Eldar and Chaos, and Imperial Guard, Tau and Necrons in later expansions. You'll have to find it yourself though.