The app is called RIFT Mobile and is currently available for the low, low price of "free." The app allows RIFT players to talk to friends and guildies whether they're in-game or also using the app -- and yes, there are push notifications if you so desire. The app also supports guild and officer chat, guild messages of the day, zone event notifications, and the guild wall.
Probably the most intriguing feature of the RIFT Mobile app is something called "Lootables" -- ways to earn actual in-game loot while using the app. The app is available in English, French and German languages and can be used only by current players of the game. A community rep on the official forums has confirmed that it's "an early release/beta" and that more official info is on the way.
[Thanks to Valotith for the tip!]