The part of Rome that continued to exist intact for such a long time after the "fall" of Rome was commonly known as the Byzantine Empire.
By the time that Rome "fell," the part of the Roman Empire that was actually ruled from Rome was called the Western Roman Empire. There was another part of the Empire, known as the Eastern Roman Empire, that was ruled from Constantinople, which was also called Byzantium. This is the city that is in what is now Turkey and is now called Istanbul
After the Western Roman Empire "fell" in 476 AD, the Eastern Roman Empire continued to exist. It came to be called the Byzantine Empire.
After the fall of the Western Roman Empire (476 AD), the Eastern Roman Empire simply went by the name the "Roman Empire". It never called itself the Byzantine Empire, that was a name made up by historians centuries later. The Empire was also known as the Empire of the Greeks to people of northern Europe. Up until 1453 AD there was one ancient Roman Empire. Starting from 800 AD there was another roman empire.... the Holy Roman Empire. That empire was not roman and it really wasn't an empire but, it did last from 800 to 1806 AD in one form or another. So, the true Roman state lasted some 2200 years and the new Roman state lasted a 1000 years.