Monument Valley is a true gem and one of the best games I have ever played on iOS & Android. It's a very simple puzzle game (there aren't complex controls) that is set in a world inspired by the art of M.C. Escher (impossible architectural sketches).
Tweetbot is an extremely popular Twitter client (a third party app for using Twitter) for iPhone and iPad. You may have already heard of it, but I use this app more than any other app on my phone so I thought it deserved a place on This Week's App!
Dark Sky is the most reliable weather app I have ever used - and I have used (and continue to use) a lot of different weather apps. The app provides users with hyper local forecasts (i.e. "Rain in 5 minutes") and does so with an extremely high degree of accuracy.
Keeping in touch with friends and family is tough, really tough. At least for me it is not too far from exercising - you know you should do it more often, but you just don't. That is why Bond exists - it's the app for keeping and maintaining bonds with those you care about.
Think way back to 2002 when Paypal was cool. Now fast forward a decade and think about how Paypal might look if it was still cool and understood mobile? That's Venmo and it is that cool and they definitely understand mobile.