Maybe the most popular turn-based game for all the times - Civilization now has a new and free variation called FreeCiv. In the game you have to develop your own empire, starting from the stone age and reaching to the space age. Your mission is to build a great civilization, to fight competitors and to develop your nation at all means.
You can download and install this game on your computer for free.
There is a online version of the game. You can find it at: play.freeciv.org . It can be played directly from your browser. Online version is single player only. Online you can play single player, but the better option is to download the game.
May your civilization grows and prosper. ;) Happy gaming. :)