This offer is set up via AmazonLocal and anyone (in the US) can participate.
Free Voucher to Purchase Select eBooks from the Amazon Kindle Store for $1
Click to "purchase" the voucher, sign in (to Amazon) and click thru a few more pages and you'll find the voucher code you need under "Your Vouchers" (link in top/right corner of the page). Sign-up for this offer expires December 18, 2012 or when the number of vouchers set aside by Amazon have been exhausted.
To use the Voucher, just click back to Amazonlocal and then on View Code next to the offer title. Click to copy the code, the on the link in step #2, to apply the code to your account (make sure you see the green "Success" message after entering the code). It works like a gift card and once applied to your account first, you can then shop from your Kindle or desktop, picking any one of the 50 qualifying titles. You should be able to gift the book to someone else (I have in the past), 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). The voucher will expire if not used toward a qualifying Kindle book purchase from Amazon.com by 11:59 p.m. PST, December 26, 2012
I don't see this offer on any of my Kindles (eInk or Fire), but it is set up just like the Fire Special Offers I've signed up for in the past (via AmazonLocal), so I wouldn't be surprised if it shows on the Special Offers page for some people.