The goal of Fallout Shelter is to ensure that the residents in said vaults remain healthy and happy. Residents can be equipped with armor, various outfits and weapons, and can undertake several activities, including procreation, through which new residents are created. You can assign residents to head out into the Wasteland to collect resources, and task them with jobs that improves their stats. During the E3 keynote, Pete Hines said:

This is inspired by other games we love, going back 30 years to Little Computer People. You'll see inspirations obviously from XCOM, Sim City, FTL, things that we really really like.

Bethesda will let you buy rewards, which come in the form of lunch boxes that contain essential resources, but other than that, Fallout Shelter is free to play. There aren't any paywall timers, and you do not need an internet connection to build and oversee vaults:

Don't worry, there are no paywall timers, you don't need an internet connection. You build something, it's instantly built. Again, no internet connection, you can play anywhere.

The game is now available for the iPhone and iPad from the link below.