It was founded by Augustus Caesar, who had assumed supreme power in Rome following the assassination of his uncle Julius Caesar. He realised that the unbridled expansionism of the Roman Republic was not sustainable, and consciously decided to keep the empire within the Rhine, the Danube and the Euphrates. His successors conquered few areas outside this, notably most of the British Isles in 43AD and Dacia and Arabia in 106.
Even holding the Rhine/Danube frontier proved hard, as the tribes to the north constantly attacked the Romans. Eventually, some of these tribes (Huns, Vandals, Visigoths, Ostrogoths) conquered large parts of the empire and deposed the last emperor, Romulus Augustulus, in 476.
The eastern part of the empire remained unconquered and continued as the Byzantine Empire, with its capital at Constantinople. This lasted nearly another thousand years, until Constantinople was conquered by the Ottoman Empire in 1453.
In Europe, the name was revived for the Holy Roman Empire.