This offer for those with a Kindle Fire with Special Offers only.
On your Kindle Fire with Special Offers (2nd Generation or HD), swipe the top menu to the left until you see Offers, then press/click in order to see the current offers. At the bottom, you should see the offer to Get a Select Music Album for $3. Click on the offer, then on the orange button to add the credit directly to your account. It's nearly instant and the next screen offers to take you to Amazon MP3 store so you can start shopping. You can also shop the Amazon MP3 store from your PC or other Android device and the credit will automatically be used on the next qualifying purchase (I prefer to browse on my PC, as you can do a "play all" on the samples).
Other than picking out an album that has a price of $10 or less, there doesn't seem to be any restriction on which albums you can choose.
Offer is good for one album and only one per account (if you have multiple KSO devices, they need to be registered to different accounts when you apply the credit to any of the accounts). The credit must be claimed by Oct 2 and used by midnight (PT) October 10.