The Byzantine Empire "began" in A.D. 476 and ended circa A.D. 1453. I put began in quotes because the Byzantine Empire din't really begin, it simply split from the Roman Empire. The Byzantine Empire was the eastern portion of the Roman Empire, and was the seat of the Empire's capitol, Constantinople, under the Emperor Constantine. When barbarians invaded and destroyed the western half of the Empire, the eastern half withdrew and defended its borders. This eastern half, now known as the Byzantine Empire, lasted another 1000 years before falling to the Muslim Empire.
The Byzantine Empire contributed greatly to the world through inventions. The Empire's inventions and innovations mostly centered around art, architecture, and religion. The artists of the Empire developed the art form know as mosaics, where they arranged colored stones to form images. The architects utilized a feature known as the onion dome, a feature often seen in modern Russian architecture. The religion of the Empire was a spin-off of Christianity, known as Greek Orthodox Christianity. This to was adopted by Russia eventually. One of the Byzantine Empire's greatest contributions however was their preservation of Greek and Roman culture. The Empire safeguarded the art, philosophy, and ideas of these two great civilizations throughout the Dark Ages.
One major contribution of the Byzantine Empire was to the realm of law. I am of course speaking of Justinian's Code. This document was an refinement and modernization of the Roman Twelve Tables. On the level of importance with the Code of Hammurabi, this document ensured an ordered and mostly just civilization for the Byzantines.
One interesting bit of information on the Byzantine Empire involves their political system. What's so special about that, one might ask. What if I told you it involved chariot races? Intrigued yet? During the reign of Justinian, the sport of chariot racing was very popular in the Byzantine Empire, although not for the reasons one might expect. You see, there were set teams that would race each time, and they were identified by color such as the Blue Team, the Green Team, etc. Now we get to the political part, the teams each represented a specific opinion or idea in an issue. The spectators would root for the teams based on their political views, thus giving the common populace a way to express their views on subjects of politics, religion, and the like.