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 Top-Rated Kindle Book for $1. 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 so you can see the list of books included in the offer. If you miss that screen, though, don't worry - a document about the offer will appear on your home page, complete with a link to the qualifying books. I thought the offer said I would also get an email, but haven't seen anything show up, which means you have to keep the Doc on your Kindle if you want a reminder about the offer until you take advantage of it, although you can shop from your desktop, using the link I've included above; I tested this and it does work, including still allowing you to send the $1 book as a gift (the full price shows all the way thru, but on the invoice summary, you will see the $1 promotional price).
I already have several of these, but there are at least a couple that look interesting; two of them also have $1.99 companion audiobooks and another two are $4.99.
Offer is good for one book 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 Sept 27 and used by midnight (PT) October 4.