This offer is only for those with a Kindle Fire with Special Offers, apparently, as I don't see it on my eInk Kindles:
10 Kindle Books for $1 Each
Click on the offer, then click on the link on the offer page sign up and you'll get an email confirming the credit has been applied to your account. This offer is good for up to 10 books (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). Sign-up for this offer expires June 29, 2013, but I would not delay, as I've often seen offers disappear from my Kindles before the end dates.
The email will contain a link to the promotion page; once you've applied the promo code from your Kindle Fire, you can shop from your Kindle, tablet or desktop; you may also gift the book to someone else (up to the 10 books limit). The full price shows all the way thru (just as with the AmazonLocal offers), but on the invoice summary, you will see the $1 promotional price. Once you have claimed the promotional code, you have until July 6, 2013 to complete your purchases.
These seem to be Amazon publishing imprint titles, once again. I have a couple of them in my TBR list, but it doesn't appear I've read any of the ones offered, this time around.