I like using Limewire for downloading individual songs.
www.limewire.com
But don't use it to download whole albums cos it will play as one file right through and not as individual songs.
To download bigger files like albums, use Bit Torrent. You need to first download the Bit Torrent client and there are loads around, but sticking with the official version is fine at
www.www.bittorrent.com
Next you have to go to a torrent indexing site to find the files.
www.newnova.org and
http://thepiratebay.org are two main ones.
Some people will say it's all illegal and stuff, but don't worry about it. It's just scaremongering. Hundreds of millions of people don't give a damn cos you have more chance of getting run over by a bus than getting caught. It's like worrying that your plane might crash everytime you fly.
Do read up about P2P though. Check this site:
www.slyck.com
Download to your hearts content, I say. Happy downloading!