Unlike in the western roman empire, most people in the byzantine empire spoke

 Greek

I might add a little detail to this.  Ever since the conquests of Alexander the Great  in the 4th century BC, the Meditarranean world in general experienced "Hellenization." Or we might call it "Greekification."  The koine (that is "common") Greek language became a common language throughout the lands where Alexander had cast his influence.  When the Romans took over those lands, Greek remained a commonly spoken language.  So Latin was the official language of the Roman empire and Romans themselves were native Latin speakers.  But Greek had become the language of international commerce.  When the New Testament was written in the early AD era, it was written in Greek for that reason.  

So, in the Western Roman Empire, Latin remained spoken, but with Greek spoken alongside it.  In the Eastern (or Byzantine) empire, Greek was predominantly spoken, while Latin remained known by educated persons, especially in government.  And Greek did become the official language of the Byzantine Empire by the time of emperor Justinian.  Justinian's famous law code, the Corpus Iurus Civilis ("Body of Civil Law") was published in both Greek and Latin editions.