It's a service where a company hosts almost all the hardware needed to play a game. They send everything to you over the net and all you need is a simple computer or a tv with a controller and an OnLive box hooked up to it. Because the processor, GPU, etc is all on their end, you don't need to have a gaming computer (or even a computer at all if you hook it up through your tv) to play a game.
Obviously this requires a flawless net connection as well as overcoming some other technical hurdles. Furthermore, if the net goes down or runs slowly you can't play any games because the games aren't actually on your computer. On the other hand, you don't need to pay massive amounts of money for a gaming computer or console, and it supposedly has some impressive networking and multiplayer functionality.
Here are a few links.
OnLive
BBC NEWS | Technology | OnLive games service 'will work'
OnLive killed the game console star? | Engadget