A capital city was the city or town that functioned as the seat of government and administrative centre of a country or region.
On Earth
Karakorum was the capital of the Mongol Empire. By 1289, the capital had moved to the Imperial Palace in Peking, leaving Karakorum abandoned. (TV: Marco Polo)
London was the capital city of England. (TV: "An Unearthly Child")
Washington was the capital city of America. (TV: The Return of Doctor Mysterio)
Brasília was the capital city of Brazil. Clyde Langer had erroneously believed it was Rio de Janeiro. (AUDIO: The Thirteenth Stone)
Sarah Jane Smith noted that Coldfire Construction's sites were "not all capital cities." (TV: Revenge of the Slitheen)
In the 2010s, the Twelfth Doctor and Nardole discovered that Harmony Shoal had established institutions in every capital city on Earth, including Washington. Though the Doctor believed all their bases were in capital cities, he was corrected by Nardole, who pointed out that New York City was not a capital city. (TV: The Return of Doctor Mysterio)
On other planets
Kaalann was the capital city of Skaro. (GAME: City of the Daleks)
Tivoli's capital city had a sign saying, "if you occupied us, you'd be home by now." (TV: Before the Flood)
Capitol City was the capital city of the human colony planet Tiestus in the 22nd century. (PROSE: Status Update)