The Roman Empire was not divided into countries. This was because: A) Countries in the modern sense of the word (nation-states) did not exist then. There were kingdoms, territories of ethnic groups named after the group (which could be regional in size and were collections of city-states of federations of tribes, rather than unified territories under one ruler) and city-states (whose size varied from a town and its surrounding countryside to region-wide territories which included other towns). B) Empires do not get divided into countries because countries are independent entities. The European colonial empires were divided into colonies, not countries.
The Roman Empire was divided into (Roman) provinces. Their number varied over time. During the Republican period there were 15 provinces. During the period of the Principate (27 BC-284 AD) the number of provinces increased to 53 due to new conquests, annexations of client kingdoms. and subdivisions of existing provinces.
In the 290s the emperor Diocletian undertook a thorough reorganisation of the administration of the empire. He subdivided the provinces into smaller units, more than doubling them and grouped them into dioceses which came under four praetorian prefectures, the largest units of the empire. There were 12 dioceses and 122 provinces.