Most often, the term Roman Empire refers to a time period, beginning with January 16, 27 B.C. (The student's Roman Empire, by John Bagnell Bury) when the title of Augustus was proposed for Octavian. This period ends with the fall of Rome, whether by fall you mean A.D. 476 (Gibbon's choice based on the last emperor in the West vacating the throne) or 1453 (when the Byzantine Roman emperor lost his power to the Turks) or some other date. The Roman Empire by this definition is divided into parts, including the beginning part, the Principate, when there was still a pretense that the government was Republican, and a later period, the Dominate, when this was no longer at issue. The Roman Empire as a whole period in Roman history is often referred to as the Imperial period.
The second meaning of the term Roman Empire is spatial. The Roman Empire began when Rome began to expand its territory during the period of the Roman Republic, especially from the mid-third century, and continued into the Imperial period, reaching its greatest extent under the Emperor Trajan. The Roman Empire came to dominate the Mediterranean and extended into Europe, Asia, and Africa.