How can I redownload itunes music?

If you didn't make a back-up of your music/movie purchases to restore from (as you are advised to do) then I'm sorry to say that the policy on music/media purchases is that you only get one download: iTunes Store: Purchased content can be downloaded only once

If you have an iPod that still has any of your purchases on it you'll find details in this article on how to transfer them to the computer: Copying iTunes Store purchases from your iPod or iPhone to a computer

Failing this, you could try contacting iTunes Store Customer Service and you might be able to persuade them to sanction a second free download. There have been occasional reports in the iTunes forum of this happening, however just be aware that they are not under any obligation to do so. If you click on this link it'll take you to the support/contact page, from there click on Music Purchases then the Lost or Missing Items option, from there click on "Email Us": Customer Service - iTunes Store Support

You might also want to check out these pages for future reference:
How to back up your media in iTunes
Back up your iTunes library by copying to an external hard drive

flundsten wrote:
I can read on the internet that it is possible to redownload music purchases

unfortunately, that information is incorrect. click here for information.

however:

Niel posted in this thead http://discussions.apple.com/thread.jspa?messageID=11029216#11029216

Niel wrote:

How come it's not the same with music purchasing?

Because Apple pays the music copyright holders a royalty each time a song, including a free single, is downloaded from the iTunes Store. You can sometimes get one free redownload request granted by using this form, but never more than one.

so, try what Niel suggested and you might be able to re-download.

for future reference, see _*The Complete Guide to Backing Up your iTunes Library*_.

good luck !

J GG