Byzantine is not actually the name of a place, in fact, it is the name of an empire, the Byzantine Empire. The term is traditionally used since the nineteenth century to refer to the Greek speaking extension of the Roman Empire which went on for a period of more than a thousand years into the Middle Ages, revolving around its capital city, Constantinople (or Byzantium.).
In particular specified contexts, it also describes the time period preceding the collapse of Western Roman Empire, also sometimes called the Eastern Roman Empire. However, there is no agreement as to the specific time period when the Eastern Roman Empire actually turned into the Byzantine Empire, the principal cause of these being that the residents of the empire merely continued to regard themselves as residing in the same Roman Empire instead of establishing a new empire.
In particular specified contexts, it also describes the time period preceding the collapse of Western Roman Empire, also sometimes called the Eastern Roman Empire. However, there is no agreement as to the specific time period when the Eastern Roman Empire actually turned into the Byzantine Empire, the principal cause of these being that the residents of the empire merely continued to regard themselves as residing in the same Roman Empire instead of establishing a new empire.