This offer is only for those with a Kindle with Special Offers (including any Kindle Touch or Kindle Keyboard that opts in to Special Offers, but not Kindle Fire with Special Offers):
Get $5 to Spend on Digital Music at Amazon
Click on the offer, then click on the link on the offer page sign up and you'll get an email with the promotion code. Sign-up for this offer expires October 7, 2012.
You'll get an email (right away), a link to the Amazon MP3 store page, a promotion code and a link to the redemption page. This one works like a gift card and you need to apply the credit to your account first, before going shopping. Once you have entered the promotional code(s), you have until October 14, 2012 to complete your purchases.
I'd recommend that you apply the credit right away, since it works on any MP3 purchase - then you can shop on the web or from your phone or Kindle Fire and it will be used automatically. The exception would be if you have the MP3 Album for $3 credit on your account. These credits don't combine well (when I tried to buy my album, it ate up all my existing credit, rather than only $3, and I had to get customer service to help straighten it out). So, use that credit first, then apply this credit to your account.
Edit to add: I found the best way to ensure that your credits are being used the way you want: send your MP3 choice to yourself as a gift. On the first page you'll see the Amazon price (ignoring any promos or credits) and you enter your email address. On the next page (the confirmation page), you'll see the exact price you pay and any promos that are being used for the purchase. By doing this, you can make sure that aren't using $8 of credit for what should be a $3 album (which is what happened to me the first time). In order to use my $3 album credit, it looks like you need to make sure you don't have any coinstar credits first.