Q: Lost old computer - can't download past purchases

Whether you can redownload music depends upon what country that you are in (and if you've changed countries since buying it and if it's still in the store). Does it show in the Purchased link under Quick Links on the right-hand side of the iTunes store homepage for redownloading ? If some music shows there nut not all then check to see if any is hidden : iTunes Store: Hiding and unhiding purchases.

Or you can copy it back from your back up of your downloads/library and add it into your new computer's iTunes via File > Add To Library

Or if you have it on your iPod then does File > Deivces > Transfer Purchases copy it back ?

I don't know whether you are using Windows or Mac, but for windows there is a program called SyncToy which you can use : Backup your iTunes for Windows library with SyncToy

Or you could just backup to an external drive : iTunes: Back up your iTunes library by copying to an external hard drive