I want to know how to add this functionality (I don’t know how it calls), when a user clicks on application in Facebook, this window is shown:

How to change traditional “Request for Permission” windows to this one?
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In your application settings in Facebook you can configure the new authorization dialog:
https://developers.facebook.com/apps/APP_ID/auth
Just follow screen instructions, you can add a description, headline, extended permissions, etc. (If you add extended permissions, a second permissions screen will be shown to the user, so the user would have to go through 2 screens before actually entering your app).