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.