If you didn't make a back-up of your purchases to restore from then I'm sorry to say that the policy on lost purchases is that you have to pay to download them again:
"Once a Product is purchased and you receive the Product, it is your responsibility not to lose, destroy, or damage the Product, and Apple shall be without liability to you in the event of any loss, destruction, or damage."
iTunes Store: Purchased content can be downloaded only once
You could try contacting the iTunes Music 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 the link at the bottom of this page you'll be able to email your query:
iTMS Customer Service
If you have an iPod that still has the purchases on it the transfer of purchased content from the iPod to authorised computers was introduced with iTunes 7. You'll find details in this article:
Copying iTunes Store purchases from your iPod or iPhone to a computer
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
