With apologies in advance to professional and amateur historians, here are a few fun "facts."
First, the nutshell version: The Roman Empire is not to be confused with the Roman Republic, except that the latter was an extension of the former and is sometimes included in a broader definition of the former. The Holy Roman Empire is not remotely related to the Roman Empire, other than perhaps through the desire of the former to be favorably associated with the latter. The Byzantine Empire is also not to be confused with the original Roman Empire, though some consider it an eastern extension of the Roman Empire. And the Byzantine Empire and the Ottoman Empire are entirely different animals, except that the latter helped to demolish the former and ended up dominating much of the same territory.
Now a humble attempt at clarification. The Roman Republic began in 509 B.C. and continued until 27 B.C., when it became the Roman Empire, which lasted until about A.D. 476. The empire ended when Rome was toppled by Odoacer, a Germanic warlord. So Roman domination of the Mediterranean basin lasted for almost 1,000 years, with roughly the first half under a republic and the second half under an emperor. Under the modern definition of "empire," it was an empire the entire time, in that it ruled over conquered nations and vast territories. What distinguished the republic from the empire was the form of government, the former being more democratically controlled and the latter ruled by an emperor (thus "empire").
Rome was the capital of the Roman Empire until A.D. 330, when the Emperor Constantine converted to Christianity and moved the capital to Byzantium, whose name he changed to Constantinople (Istanbul today). When Odoacer sacked Rome in 476, the Roman Empire died -- except it didn't altogether, only its western part. It could be said to have continued in its eastern part, as the Byzantine Empire began in its wake and continued until about 1453, when the Ottomans conquered Constantinople. So speaking very broadly, you could say that a Roman empire, in some form, existed for some 2,000 years, from 509 B.C. to A.D. 1453, as follows: Roman Republic (509 B.C.-27 B.C.), Roman Empire (27 B.C.-A.D. 476) and then the Byzantine Empire, or "Eastern Roman Empire" (partially overlapping with Roman Empire, A.D. 330-1453). The Ottoman Empire, by the way, continued until the early 1900s.
These geographical empires are not to be confused with the church hierarchies, which is not to say there isn't significant overlap and interplay between the two. Concerning the church, we must now distinguish between the churches resulting from the Great Schism of 1054 (also called the East-West Schism or sometimes even the Schism of the East), when the church in Byzantium split from the church in Rome to form the Eastern Orthodox Church. Henceforth, there would be the Roman Catholics in the west and the Eastern Orthodox Church in the east.
This Great Schism is not to be confused with the Great Schism of 1378 (also called the Schism of the West), which involved only the western church -- i.e., the Roman Catholic Church. When Pope Gregory XI died that year, there was a dispute between the French and Romans as to whether his successor would be French or Roman and whether he would serve in Avignon or Rome. The chaos even resulted in three competing popes for a while, but the schism only lasted 40 years, until a new conclave elected Cardinal Oddone Colonna, from a Roman family, as Pope Martin V in 1417.
Finally, we should not confuse the Roman Empire, in any of its manifestations, with the Holy Roman Empire, which was more Germanic than Roman. It was hardly an empire; it was more a loose confederation of states located in central Europe, not the Mediterranean basin, which the Roman Empire dominated. Its connection to Rome is that the pope bestowed the title of emperor on Charlemagne in 800, though Otto I, crowned emperor in 962, is considered to be the first formal Holy Roman emperor. The Holy Roman Empire continued, with ebbs and flows, until 1806, when Napoleon dissolved it. You might better remember the Germanic nature of the Holy Roman Empire if you recall that it was considered the First Reich. The Second Reich began after the Franco-Prussian War (1870-71) under William I and his chancellor, Otto von Bismarck.
Confused yet? Well, just be glad I don't delve into this paradox: Many historians consider the absolutist emperor Napoleon Bonaparte to be a great liberal reformer.