The Byzantine Empire is the modern historical name given to the Roman Empire's eastern half, centred on Constantinople, but its own inhabitants throughout its history simply called themselves Romans and their state the Roman Empire, without drawing the sharp break modern terminology suggests. While the western Roman Empire, the one most popularly associated with "ancient Rome," collapsed in the fifth century, this eastern continuation persisted for roughly another thousand years, finally falling to Ottoman conquest only in 1453.
The empire's eastern half kept functioning long after the western half collapsed
The Roman Empire had already been administratively divided into western and eastern halves before the western half's collapse, and this eastern half, centred on the wealthy, strategically positioned city of Constantinople, continued functioning as a coherent state with its own emperors, administration and military long after the western Roman state had fragmented under pressure from various invading groups. Rather than treating this as some entirely new, separate empire, its own inhabitants and government continued to describe it in direct continuity with Rome, seeing themselves as the Roman Empire's rightful, ongoing continuation rather than as its historical successor.
Modern historians gave it a separate name the empire never used for itself
The term "Byzantine Empire" itself is a later historical convention, derived from Byzantium, the original Greek name for the site Constantinople was built on, and it wasn't a label the empire's own inhabitants used for themselves at any point during its roughly thousand-year existence following the western collapse. Historians adopted the term retrospectively, partly to distinguish this later, culturally distinct, Greek-speaking and Orthodox Christian phase of Roman history from the earlier, more familiar Latin-speaking ancient Rome, even though the empire's own self-understanding maintained direct, unbroken continuity with Rome throughout.
What we're still unsure about
That the Byzantine Empire represented a direct political and administrative continuation of the Roman state, and that "Byzantine" itself is a later historiographical label rather than a term the empire used for itself, are well established, extensively documented historical facts. What's more genuinely a matter of ongoing historical debate is exactly where to draw the meaningful dividing line between "ancient Rome" and "Byzantium" for analytical purposes, since the transition was gradual rather than a single sharp break, and historians continue to propose and debate different candidate turning points, changes in official language, religion, or territorial control, without full agreement on which single marker most usefully separates the two conventional historical periods.
This sits inside The Byzantine Empire, one of eight topics in Medieval History, one of seven domains in History, one of seventeen subjects the app can quiz you on.