PlayStation App is an official application developed by Sony that allows you to bring the PlayStation experience (whatever that means) everywhere on your Android device.
Thanks to this PlayStation App, you can see what your friends are playing at any given moment, compare their achievements with your own, and check your profile and recent PlayStation Network activity.
You can also, naturally, chat with all your friends, receive notifications, game alerts, and invitations, and use your mobile device like a keyboard when you're using your PlayStation 4 (it's much more comfortable than using the controller).
Even better, you can use the app to navigate the PlayStation Store, choosing games and popular add-ons, which are then ready to play on your console once you get back home. Basically, you can use your smartphone to start downloading an update or game to your console.
PlayStation App is an intriguing application for any PlayStation 4 owner, since it gives you access to a good amount of interesting features.