The Byzantine Empire was the Eastern Roman Empire, its capital being Constantinople (modern-day Istanbul, Turkey). It is what remained after the fall of the Western Roman Empire, and because it lies on the eastern side of the Ancient Roman Empire the official language was Greek, and the empire adopted Orthodox Christianity, distinguishing it from the Latin west.
The capital of Constantinople gave the Byzantine Empire significant strategic advantage, as it was right on the trade routes between Europe and Asia as well as the Mediterranean and Black Seas.
The division of the empire into east and west often also split the administration of the empire, as was the case under Theodosius I when he left the east to one son and the west to another. This allowed for the east to continue to thrive and defend the empire despite the continuing downfall of the west. Due to its proximity to expanding Germanic nations and overall weakness, the western empire eventually disintegrated altogether in the late 5th century.
The eastern side, due to its proximity to the Huns and later the Sassanids and Persia in the east, often had to pay tribute to these tribes but remained strong. Byzantine borders expanded to their largest after reclaiming some of the western Roman territory during the rule of Justinian I.
The Germanic invaders to the west and the Persians and Arab Caliphate to the east persisted, however, and after Justinian I died the Byzantine Empire began to lose territory on both fronts. Eventually, in 626, Constantinople fell to a combination of Sassanid Persia, Avar, and the Slavic forces, its strategic location eventually leading to its downfall.
The Arabs continued to raid and wage war against Byzantium, and eventually wars against the Bulgarian Empire, now their neighbors to the west, began as well. By the 11th century they had effectively conquered Bulgaria and annexed parts of Armenia and Southern Italy, and reconquered the important city Antioch. After these geographical gains Constantinople flourished, becoming the wealthiest city in Europe.
But the gains meant that Byzantium encompassed Catholic and Orthodox Christian territories, causing a split and crisis within the empire. The gain of southern Italy put them next to the advancing Normans, who took advantage of this schism. Meanwhile, the Seljuq Turks from the east began to explore, and advanced through Armenia, nearing Constantinople.
The empire continued to struggle with the east-west schism, unable to defend its whole without uniting. The west had now achieved some stability, but the east was losing the the Turks, and they asked the Pope of the west for help. He agreed, and thus began the crusades of the 12th century.
Tensions got stronger and the crusaders began to advance over the eastern Byzantine empire, eventually attacking Constantinople in 1204. This split the empire further. Civil war eventually broke out in the early 14th century, allowing Serbia to take much of the empire. After that, an earthquake devastated the fort in Gallipoli, allowing the Ottomans to begin overtaking Europe. The Ottomans defeated the Serbians while Byzantium was busy in civil war. The Byzantine emperors appealed to the Pope in the west again, but this time they didn't receive help. Constantinople finally fell to the Ottomans on May 29, 1453. Their strategic location between east and west caused a perpetual split that eventually brought their downfall.