App of the Week is an iTunes App Store weekly affair where one selected iOS app is prominently featured for free for a week as “App of the Week”. You can follow App Store’s official Tweeter account @AppStore for news when an app is featured, or just search for #FreeAppoftheWeek hashtag.
On week when the app isn’t apparently relevant, just download the free app and then delete it. You can always download the app again at later time from the App Store app at Updates > Purchased. Free app downloaded is considered your purchase, and is free for your download even when the promotion is over.