While I'm not exactly the world's biggest board game enthusiast, Tigris & Euphrates is one even I've heard of: the much-lauded strategy game has transcended its cardboard and tile roots and is now on Android. The proto-historic pastime designed by Dr. Reiner Knizia has been around over 15 years now, and this Android version is an official port.
Right now, only hot seat multiplayer is supported, but the developer promises online multiplayer is 'coming soon.' Three game modes are available: solo, pass and play, and hotseat. A full tutorial is included, along with the official rule book. Head to the Play Store to get it now.