It is a custom Facebook App. Called a "Fan Gate" or "Like Gate".
You can check out pagemodo.com, it looks like it will build the "like" part for you. It's what creates the tab named "View Your Photos" on their facebook page.
It looks like the one for Booth Revolution is a customized (probably paid for) version of that, sans the usual advertisement at the bottom.
It is then using a link back to the boothrevolution site at boothrevolution.zenfolio.com
This then is using AJAX/JSON and .Net for password authentication.
zenfolio.com seems to have a trial period to try out the service, or you'll have to pay for it.
If you host your own photos then you'll just have to password protect your pages on your own.
If you really, really want to go your own way then this article seems to have a good start in the right direction on building the Fan Gate part: http://www.makeuseof.com/tag/like-to-unlock-how-to-build-a-basic-facebook-fan-gate-without-paying-for-hosting/