Currently there is a lot of fuss about Wordfeud, a turn based Scrabble(tm) like game, on the tablets and smartphones. Turn based games are already as old as there is connection over the mail between people. There are played hundreds, thousands of chess games by people sending each other a postcard with the next move. This is called "correspondence chess". When internet became more and more popular, people started to play through email. I, for example, started to play with the IECG. Which is nowadays defunct, because server based playing took over the world.
Not only for chess, but also for other games like Go, Connect6, Draughts, Checkers, Einstein Wuerfelt Nicht, etc. And the latest move is bringing these server applications to the smart mobile phones. For me there are two servers where I play my games: the Lechenicher Chess Server and LittleGolem. The Lechenicher is the follow up for the IECG. I'm playing chess over there almost since the start of the server. With great joy and fun. On LittleGolem you can choose what kind of game you want to play. Not only chess, but also other games are possible. I'm playing mainly Einstein Wuerfelt Nicht, Connect6 and Havannah on LG.
Have a look at both. They are free, but you are allowed to pay a little money for keeping the services on the net. And you get some extra priviliges if you pay a little money.