Fan of the Week is an add-on for Facebook Pages. It makes fans use your pages more actively, encouraging them to like your page, its posts and to post on the wall themselves if you would like to encourage that. Because of Facebook's EdgeRank, when fans interact with your pages more, your page updates will also be more likely to appear on their news feed. So in short, you should really go add it to your page right now.
Here is one example of how people are using this add-on on their pages. Club D'Ville is a nightclub in Ireland. They have been using Fan of the Week to not only pick a fan to get the honorary title, but also give them free entrance tickets to the club. Every week the app automatically picks a fan and announces them as that week's fan. The announcement is customizable, as you can see here:
Below you can see what the add-on page tab looks like. It is also completely customizable. Fans are picked based on not just random chance, but also on how much they interacted with the page. Each page interaction is like a lottery ticket that gives the fan a chance to get picked.
If you have a Facebook page and think Fan of the Week would be a nice addition to it, you can add it to your page here.