Here's a snippet from their website about the name:
The name Pandora means "all gifted" in Greek. In ancient Greek mythology, Pandora received many gifts from the Gods, including the gift of music, from Apollo. She was also, as we all know, very curious. Unlike those Gods of old, however, we celebrate that virtue and have made it our mission to reward the musically curious among us with a never-ending experience of music discovery.
So, the basic idea is that you type in an artist or song that you like, and Pandora searches the "Music Genome Project". While you may not instantly hear that song or artist, you WILL hear a song with similar qualities. Then you have the option of rating it thumbs up, or thumbs down, and Pandora will continue playing songs, using your feedback to create your personal station.
I have an Alison Krauss station that is fantastic (With *mostly* Alison Krauss music, although I've heard JT, EmmyLou Harris, etc... and some new names that I'm enjoying getting to know).
I created a Josh Groban station for mellow music to school to, A "Boppin' Praise" station with the likes of Chris Rice and Casting Crowns, and a couple of folk stations. I have yet to create a classical station or two, but that's a thought, and I bet that I'll get a Christmas station going in the later fall!
I really enjoy being able to hear new music that is in a style that I appreciate. I think this is a great thing, and my kids are loving it too.
Check it out!