When you sign up to your chosen site, you will usually be given instructions on how to download software to play music before you even download a song. The software for each provider does vary but here are a few examples:
iTunes
If you own an iPod and you are downloading music from the iTunes Music Store then you already have all the software you need to play the downloaded songs on your computer
Napster, MyCokeMusic and many others
To play songs from Napster, MyCokeMusic and a number of other sites you will need a piece of software called Microsoft Media Player, which comes with most recent PCs and is downloadable for free from Microsoft if you need to get it.