The term Byzantine comes from Byzantium, the original name of Constantinople before it became the capital of the Roman Empire. The term came into general use in Western Europe during the sixteenth to nineteenth centuries to describe the Roman Empire after the fall of the western half of the Roman Empire in 476.
After Constantine I moved the capital of the Roman Empire from Nicomedia to Byzantium in 330, renaming it Nova Roma, the empire continued to be referred to by its inhabitants as the Roman Empire (Greek: Βασιλεία Ῥωμαίων) or Romania (Greek: Ῥωμανία). After Constantine's death in 337, the city was again renamed, to Constantinople. The line of emperors continued in an unbroken succession and preserved its Greco-Roman legal and cultural traditions. After the fall of Rome and the lost of the western part of the empire, the empire began to take on a new character, becoming more Hellenic in culture. By the seventh century, Emperor Heraclius recognized Greek as the official language.
In the meantime, western Europeans referred to the eastern empire as Imperium Graecorum (Empire of the Greeks) and began to claim the inheritance of Rome for the West, particularly by the time of Charlemagne (Charles the Great) who was crowned Imperator Augustus, a preferred title of Roman emperors, by Pope Leo III of Rome on December 25, 800, as a rival of the Roman emperor in Constantinople. Other sources note the title as Imperator Romanorum (Emperor of the Romans). Thereafter, the Popes of Rome and western European rulers preferred to refer to the "Roman Empire" in regard to Charlemagne and his successors (i.e., the Holy Roman Empire) .
Use of the term "Byzantine" to refer to the area of the eastern Mediterranean Sea ruled through the capital Constantinople began late with its use by the German historian Hieronymus Wolf in 1557 in his work Corpus Historiae Byzantinae. Later publications (Byzantine du Louvre (Corpus Scriptorum Historiae Byzantinae)) in 1648, and Du Cange's Historia Byzantina in 1680) further popularized the use of the term among French authors. Then, in the nineteenth century, the term "Byzantine" came into general use among western Europeans with the birth of modern Greece.
Thus, the term "Byzantine" came to be used by the western Europeans, and especially by the Roman Catholic Church, as an appropriate way of designating the Eastern half of the ancient Roman Empire that survived in the east and the Christian Churches, under the Bishop of Rome, using Orthodox rites.