There were at least three different Roman Empires before about 1500, all centered in different places. To make it more confusing, they all asserted they were the same Roman Empire.
Let's start with the original Roman Empire, the territory conquered in (roughly) the five centuries BC, which ended up including the whole Mediterranean basin and Europe west of the Rhine. Its capital was Rome, and from the first century BC it was ruled by emperors, absolute rulers who originally were assumed to be semi-divine.
This Empire split in the first half of the fourth century, when the Emperor Constantine moved his capital to Constantinople (the modern city of Istanbul), in Greek-speaking Byzantium (now Turkey). (I wonder why it's named Constantinople?) (See more here on the "fall" of the Roman Empire.)
Constantine also converted to Christianity, a religion that soon became official throughout the Empire. Though no longer semi-divine, the emperors continued to be absolute rulers.
From the early fourth century to the late fifth, the real capital of the Empire was Constantinople, but there were still sometimes co-emperors headquartered in the city of Rome. The last of these western co-emperors was killed at the orders of the Byzantine emperor, leading to over three centuries of just one Roman emperor. These Byzantine Roman emperors indeed continued to rule until 1453, when the Turks finally took Constantinople.
Meanwhile, back in the west, the Frankish king Charlemagne was crowned emperor by the pope in the year 800. (See more here on Charlemagne.) Now there were two emperors, one with his capital at Aachen, near what is now the French-German border, and the other in Constantinople. Both called themselves Roman Emperors. Neither was in Rome.
Charlemagne's empire, which was originally (more or less) the territory that now corresponds to France, Italy, and western Germany, became by the eleventh century just Germany and Italy, as France went its own way. German kings would be elected by the German princes, then cross the Alps, beat up some Italians, and be crowned Roman emperor by the pope. (See more here on medieval Italy.) Since 800, it was very clear that no western king could call himself emperor unless the pope crowned him.
As you can doubtless imagine, there was a great deal of hard feeling between emperors and popes, who felt that the other guy ought to do what he wanted. The Investiture Controversy (on which see more here) was a knock-down, drag-out fight that continued on and off from the late eleventh century to the late twelfth. At one interlude during this Controversy, the German emperor decided that not only was he Roman emperor, he was Holy Roman Emperor. (Take that, pope!) The title sticks. Indeed, to avoid confusion, a lot of historians refer to the "Holy Roman Empire" as starting with Charlemagne, to keep it distinct from either the original Roman empire of antiquity or the Byzantine Roman empire.
In the late Middle Ages, the German princes, deciding that if they elected someone king that was good enough, declared that one could be Holy Roman Emperor without the bother of crossing the Alps and beating up Italians in order to be crowned at Rome. The Holy Roman Empire eventually was joined by marriage to the Spanish Empire of Ferdinand and Isabella, busy conquering the Mediterranean and the New World, but that takes us into the post-medieval period.
To reiterate, there's the original Roman Empire, centered at Rome, until the early fourth century. There's the Byzantine Roman Empire, centered at Constantinople, from the fourth century until 1453. And there's the Holy Roman Empire, not called "holy" until the twelfth century, but really starting with Charlemagne and persisting (with majors changes) until the end of the Middle Ages. All clear now?