The term translatio imperii, Latin for "transfer of rule", typically refers to the passing of the crown of the emperor to the Holy Roman Empire when, on December 25, 800, Charlemagne, king of the Franks, and then on February 2, 962, Otto I the Great, king of the East Franks, was crowned emperor by the pope in Rome.
It was considered to have started with the Babylonians, then passed to the Persians and finally to the Macedonians under Alexander the Great. The culmination of the translatio is with the Romans. The choice was not arbitrary as these empires were figuratively mentioned in the Book of Daniel as four beasts coming out of the sea, or, alternatively, as four parts to a statue.
The Church having given the Imperium to Charlemagne from Byzantium, the seat of the Roman Empire in those days, it eventually landed in the Holy Roman Empire (and this explains the Roman component of the Empire's name). The Holy Roman Emperors thus thought of themselves as being in direct succession of the ancient Roman emperors and were bolstered in their claim, specifically against the French monarchs who might also claim lineage from Charlemagne, by papal crownings.