The Byzantines called themselves Romaioi, which is Greek for Romans (they spoke Greek). Constantinople is an anglicized version of the Byzantine name for that city which is Konstantinopolis (the polis - meaning city in Greek - of Constantine the Great, a Roman emperor who made it the East Roman capital). The name Byzantine Empire derives from an earlier name of the city, namely Byzantium or Byzantion. Byzantine Empire is a later name for the Empire, they themselves called it the Basileia Romaion, which means Roman Empire.

Can't tell you much about other peoples' names for them though other than that the name Konstantinopolis stuck with the Ottomans as well in the form of Konstantiniyye, and the Norsemen called it Miklagarðr or Miklagard (roughly great city, from mikill which is Old Norse for great and garðr which is roughly equivalent to English dwelling place or town - cognate with English yard - see also Asgard for example which is the home of the gods, the Aesir, in Norse mythology).

Adding to /u/QueenAtziri's answer, their Arab neighbors (and Persian neighbors) referred to them as Romans as well, as did the Turks who invaded Anatolia later. Indeed, the Seljuk Sultante of Rum is named after the fact that it was established in 'Roman' territory.

To the West, they would either be called the Romans/Roman Empire, or the Greek Empire/Empire of the Greeks. For example, Pope Urban refers to them as 'Romania' in the 1095 speech he made at the Council of Clermont according to Fulcher of Chartres, a reference to their status as Romans, not as modern Romanians. As we progress closer to modernity, the later title would become more common when refering to the Empire in Western Europe.