Its actually a slightly tricky question. I think that the best answer is that the Byzantine Empire is essentially the successor state to the Roman Empire. Emperor Constantine, feeling the Empire was too large to govern, split the empire into East and West. After the Roman Empire fell, the remnant, the Eastern Empire came to be known as the Byzantine Empire. The Byzantine emperors called themselves Emperors of the Romans.
**Actually the answer above is a bit off the mark. Constantine was the Emperor who reunited the Empire. Diocletian was the one who split it into western and eastern sections. Diocletian was also the one who set up a Tetrarchy (meaning there were four rulers). Each part of the Empire had an Augustus, or senior Emperor, and a Caesar, or junior Emperor. Diocletian rules in the east, Maximian took the west and ruled from Mediolanum (now known as Milan). Galerius was placed as Caesar in the Balkans, and Constantius Chlorus (Constantine's father) was given the area Gaul and Britain to rule.
Different historians place the "beginning" of the Byzantine Empire at different dates according to which event they consider more important. There are 3 candidates for the beginning of the Empire:
1. 285 AD, when Diocletian split the Empire.
2. 324 AD, when Constantine started building the new capital.
3. 330 AD, when this new capital (Constantinople) was dedicated.
One of the best books on The Byzantine Empire is a 3-volume set written by John Julius Norwich:
Volume I - Byzantium: The Early Years
Volume II - Byzantium: The Apogee
Volume III - Byzantium: The Decline and Fall
There are many others, but this series should be an excellent introduction.