The Inbox is the place to find all of your notifications, friend invites, gifts and rewards! These can be sorted by tapping the blue bar to cycle through different filters.  When you receive an item in your Inbox you will see a number near this icon. 
Have too many items in your Inbox? You can click Claim All to claim everything you have in one tap. 
Gifting: You can gift your friends gift boxes by tapping Social, then tapping the Friends Button. From there you will be able to send gifts!  Gifts cannot be requested, they can only be sent.