Explain the origin of the Byzantine Empire.

The Byzantine Empire began as a part of the Roman Empire. 
Then, over time, the Byzantine Empire gradually split away from the Roman Empire,
developing its own culture and institutions.

The Byzantine
Empire can be said to have started in 286 AD when the Emperor Diocletian split the Roman
Empire up into four regions for administrative purposes.  This sort of started the idea
of a divided empire.

Later, in 330, Constantine founded the
city of Constantinople (later to be called Byzantium) and set it up as the capital of
the whole Roman Empire.  After the death of the Emperor Theodosius in 395 AD, the empire
was formally split between his two sons -- one ruling from Rome and one from
Constainople.

Over time, the Eastern Empire became more
Greek and therefore more distinct from the Western Empire.  When the Western Roman
Empire fell, only the Eastern Empire was left and it came to be known as the Byzantine
Empire.