A Captured Town is any town who's controlling civilization is no longer the civilization that founded the town. When a town is captured, it retains all of its management (aka the Mayors and Assistants are the same) but must deal with the government of the owning civilization and pay their taxes.
How Do You Capture A Town?
To capture a town, destroy all of the Town Hall's control blocks. If the town is the civilization's capitol, you must destroy all of the Capitol's control blocks. If you capture a civilization's capitol, you capture the entire civilization instead of just the town. See Captured Civilizations. You can also use a War Cannon to damage a town hall or capitol and do damage to the control blocks.
If the mayors of a captured town 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 town is now 100% owned by the controlling civilization and behaves exactly as if the town had been founded by the owning civilization.
If the leaders of the owning civilization choose to do so, they can liberate your town using Diplomacy.
Captured Town Unhappiness
Each captured town provides 2 unhappiness points to every town in the owning civilization.