The Western Roman Empire fell in 476 AD, with the last emperor Romulus Augustulus deposed by the barbarian king Odoacer. At the same time, the Eastern half of the Empire, Byzantium, whose capital was Constantinople, continued for almost a thousand years before being sacked by the Ottomans in 1453. So why did the East survive while the West did not?
One of the main reasons was the mass migration of Germanic tribes into the West. A large number of such tribes had been formally Romanized in the process, speaking Latin and fighting against rival tribes as part of the Roman army, where they gained a fearsome reputation as warriors. In return for their military service, they demanded their fair share of land, but for one reason or another successive emperors were either unwilling or unable to grant it to them. It was the deep, smoldering resentment caused by land hunger that was the primary cause of the many barbarian invasions, including the one that finished off the Western Roman Empire once and for all.
In the East, however, the situation was completely different. The Byzantine Emperors had much greater control over their military forces than their counterparts in the West, who tended to rely on the loyalty of Germanic generals out in the field. As we've seen in the case of Odoacer, such loyalty was not always forthcoming, not least because these generals were powerful men in their own right, with vast legions to command. If they didn't get what they wanted from the Emperor, they were always ready to descend upon Rome and take it for themselves.
The geographical separation of the Eastern half of the Empire from Germanic lands saved Constantinople from a similar fate to Rome. Furthermore, Byzantium only had to defend two active borders from potential attack, placing it in a much more strategically secure position than the West, which had to contend with aggressive tribal coalitions all along the full length of its vast northwestern frontier.