Anything that is internet capable is the short answer.
There are not that many web sites available for downloading legal copies of music tracks, I can think of two, Spotify and iTunes to begin with, and we can't discuss any illegal options.
But that makes no difference to what you use. Tablet, Notebook, Laptop, Desktop, Mac, Windows, Linux, iPad, even iPhone and other smart phones will do it.
With such broad requirements your options are huge.
Mark
Hi, what would be a good spec for downloading music and using the internet a lot, but not playing games?

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic