Choosing the "Raze City" option at any time after you have successfully occupied a city (except for a former capital or a city-state) will set fire to it, reducing its Population by 1 per turn. When the city's Population reaches zero, it will be removed from the map and replaced by City Ruins. They are also the result of using nuclear weaponry on cities with little defense and hit points.
In Brave New World, upon the discovery of Archaeology, City Ruins can be turned into an Archaeological Dig site that can be converted into a Landmark or dug for cultural Artifacts. Depending on what is chosen, this will produce either Culture or Tourism.
City Ruins are the remnants of cities destroyed in the game. They are of no particular value in themselves, but they often sit atop valuable locations. A worker may clean up a City Ruin or simply replace it with another improvement.