Byzantium was an ancient city, and later the centre of the Byzantine Empire. Constantine the Great gave it a new name, Constantinople, and today the same place is called Istanbul, in Turkey. When history books say Byzantium, they mean that old city, or sometimes the whole world that city ruled. It is a proper name, so we write it with a capital B. You will meet it in history class more than in daily talk. The related word Byzantine can mean “from that empire,” and people also stretch it to mean rules or plans that are twisting and too complicated — though that stretch is about the adjective, not the city name itself.