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 the Kindle Fire with Special Offers):
Get 1 of 10 Top-Rated Kindle Books for $1
Click on the offer, then click on the link on the offer page sign up and you'll get an email(right away) with the promotion code. This 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 before you sign up to get the promotion code emails; every Kindle on an account gets flagged when you sign up for the special offer and they will all get the same code for that promotion from then on). Sign-up for this offer expires October 14, 2012.
The email will contain a link to the promotion page. This one works like a gift card and you need to apply the credit to your account first, before going shopping. You can shop from your Kindle or desktop, after you have applied the promotion code and can gift the book to someone else, if there isn't one you want yourself (the full price shows all the way thru, but on the invoice summary, you will see the $1 promotional price). Once you have entered the promotional code, you have until October 21, 2012 to complete your purchases.
Other than the dates (and getting a code via email, rather than a credit via AmazonLocal), this is the exact same offer that those with a Kindle Fire with Special Offers received last month.