They have a “Play” button for each song. When one song’s play button is pressed, any other song playing is stopped and the new one starts (there are surely many sites with similar features, this is just an example)
I tried looking at the page source, but after a quick perusal I can’t figure out what they are using to get the desired effect.
Are they using Flash with a Javascript wrapper? Something else?
So when the user clicks the Play button, it sends the song ID and url to the radioplayer.js, and starts to play the song. Your browser most likely has a pre-installed plugin to handle the audio format.