It's an iPhone and iPad app that lets your device connect to the Tor network, which reroutes your web traffic through various servers before ultimately landing on your screen. You stay completely anonymized.
Onion Browser is simply a mobile browser that knows how to navigate all this rerouting, getting at this is a layer of the Internet that conventional browsers like Chrome and Firefox can't access.
If you want to take advantage of Tor on your phone, you'll need this app. It gets your phone connected to the same secure, anonymous network. Your browsing habits aren't recorded and you remain unidentifiable.