A Captured Civilization is a civilization who's capitol has been captured by a controlling civilization. When a civilization is captured, it retains all of its management (aka the Leaders and Advisers are the same) but must deal with the government of the owning civilization and pay their taxes.
How Do I Capture A Civilization?
If the leaders of a captured civilization wish to permanently join the owning civilization, they can issue a capitulation request to the leaders of the captured civilization. When the leader accepts, the civilization and every town in it is now 100% owned by the controlling civilization and behaves exactly as if the all of the captured civ's towns had been founded by the owning civilization.
Captured Civilizations have the option of starting a Revolution to become free again.
If the conquering civilization chooses to do so, they may liberate your civilization, returning full control back to you. Only the owner of your old capitol town can liberate your entire civilization, and only towns under his control can be liberated. Other towns owned by other civilizations must be liberated by their owners individually. See the diplomacy page on liberation for more information.