Short answers, pulled from the story.
The Byzantine Empire, also known as the Eastern Roman Empire, was the continuation of the Roman Empire centred on Constantinople during late antiquity and the Middle Ages. It survived the fall of the Western Roman Empire in the 5th century AD and endured until the fall of Constantinople in 1453.
The term Byzantine Empire was coined only after the empire fell. Its citizens used the term Roman Empire and called themselves Romans, in Greek Romaioi, while the adjective Byzantine derives from Byzantion, the Greek settlement on which Constantinople was built.
The Byzantine Empire reached its greatest extent under Justinian I, who briefly reconquered much of Italy and the western Mediterranean coast after his accession in 527. His general Belisarius subjugated the Vandal Kingdom in late 533, and the Ostrogothic Kingdom mostly ended in 554.
Sultan Mehmed II resolved to capture Constantinople in 1452 and laid siege early the following year. On the 29th of May 1453 the city was captured, the last emperor Constantine XI died in battle, and the Byzantine Empire ended.
The Fourth Crusade sacked Constantinople in April 1204, ransacking wealth the city had accumulated over nine centuries. The territories fragmented into the Latin Empire and competing Greek successor states, and although Michael VIII recaptured Constantinople in 1261, the reconstituted empire wielded only regional power.
The Byzantine Empire transmitted classical knowledge to the Islamic world and Renaissance Italy, influenced the civil law traditions of continental Europe and beyond through its legal codes, and is credited with developing the Glagolitic alphabet, which later evolved into the Cyrillic script and Old Church Slavonic. It also pioneered the hospital as an institution offering the possibility of a cure.