Chess Presso is a great app to play chess on your android device. You can play with a friend or a random person from around the world.
Gameplay is really simple. Tap the piece you would like to move, then tap where you want it to go.
The built in chat feature allows you to chat with the person you are playing.
Settings can be configured to allow for notification when it is your turn.
There is a free, ad support version. The paid version costs $4.03.