Seems like everyone has happened upon this at the same time! Not sure how long the service has been running but I just came across Pandora via TechCrunch.com. Pandora asks for an artist or song that you like and then, based on your choice and with the help of the Music Genome Project, plays a selection of music it thinks you will like. You can rate the music Pandora selects to better fine tune your personal "Station" and also listen to stations that other users have created.
While I've used services like this before such as Yahoo's launch.com which is available as part of Yahoo Messenger but Pandora just "Feels" nicer and seems to have more smarts. The Flash based player has some very nice touches and is easy and intuative to use, and the quality of the music matches seems pretty reliable. Nic did mention however, that if your taste's are slightly more obscure (as his can be) the quality of matches offered up does degrade a little, but I guess as the project grows and matures that this will most probably be addressed.