When Mohammed died in 632 CE, Islam prevailed only in an the Arabian desert area roughly aligned with modern Saudi Arabia, the United Arab Emirates, Oman and Yemen.
A short generation later, in 655 CE, the Islamic empire had conquered all of the Byzantine Empire (which was the successor to the Eastern Roman Empire after 395 CE) outside Turkey (the called Asia Minor), Greece, the Balkans, and patches of Italy that Justinian had reclaimed. The Islamic empire had also secured Armenia and Persia (modern day Iran stretching into modern day Afghanistan).
The six year civil war that divided in Islamic world into Sunnis (who supported the Umayyads Caliphs) and the Shi'ites who followed Mohammed's son-in-law followed.
The Umayyads (a.k.a. Omayyads) prevailed in 661 CE and continued the Islamic Empire's dramatic spread until their dynasty ended in 750 CE.
In the West the Islamic Empire first defeated successors to the Vandals of North Africa in what had previously been part of the Western Roman Empire based out of Carthage in what is now Tunisia. Then, they defeated the Kingdom established by the Visigoths in what is now Spain, except the Basque territory which had been autonomous since around the time that the Western Roman empire fell in 476 CE. The small makeshift Muslim army that had reached so far afield was turned back by Charles Martel (a proto-French Frank), in 732 CE. This would prove to be the high water mark of the Islamic Empire in the West.
Iberia (the Peninsula that is now Spain and Portugal) would gradually be reclaimed by European Christians over the centuries in what is known as the Reconquest of Spain. Barcelona was under Christian rule by 1000 CE. The Northern half of Iberia was under Christian rule by 1100 CE. Cordova the original capital of the Caliphate that ruled Iberia fell to Castile in 1236 CE. By 1300 CE, the Moors were limited to Granada, a small Southern province Southeast of Seville. Granada fell in 1492. North Africa remains Muslim today.
In the East, by 750 CE, the Islamic Empire had secured most of modern day Pakistan and started to encroach upon the Persian fringe of Asia Minor. By 751, a new Abbasid dynasty of Caliphs defeated a Chinese army in Central Asia. Soon after, the empire had made major inroads in the Indian subcontinent, Indonesia and China, to almost every place that Islam is found today in Asia.
The First Crusade from about 1095 CE to 1099 CE established Crusader states in the Levant and Southeast Asia Minor, but this was short lived. The Cruader state of Edessa in Southeast Asia Minor fell in 1144. Islamic prince Saladin reclaimed Jerusalem in 1187. Jersusalem remained in one or another Islamic successor state until Israel was formed (arguably earlier, when the Ottoman Empire was parceled out after World War I).
The Ottoman Empire (the largest successor to the core Islamic empire) invaded the Balkans in 1354 and ultimately conquered almost all of Byzantine Europe. The remnant Byzantine empire breathed its last gasp in the fall of Constantinople in 1453. Some of this territory in central Europe was relinquished about two hundred years later, but most remained under Ottoman control until after World War I.