Facebook Signin in a mobile App (backed by a REST API)

I have done some researching but I haven't find a clean explanation of how it all work.

I would like to know how does applicationa like Instagram, Wordcrack (online game), AirBnb and many other app have tier "Login with Facebook" done?

I currently work on the Java REST services that backups up an iOS app. If some one already has done a similar things in their app/server I would really appreciate some help

I'm not looking for ready to use code answer but more of a general idea of the architecture to achieve this.

thank you!