Final answer:
The part of the Church over which a bishop has authority is known as a diocese.
Step-by-step explanation:
The part of the Church, a geographical region, over which a bishop has authority, is called a diocese. Each diocese generally follows the boundaries of the older Roman Empire's imperial administration. This organization of the Church hierarchy has historical roots that date back to when Christianity moved from being an illegal cult to the official religion of the Roman Empire. The division into dioceses was a method of managing the larger Christian community effectively.
In the history of the Church, the role of the bishop has always been significant. For instance, the early Christians saw that bishops were important in instructing people in Christian doctrine and running charity networks. With the rise of the Roman Church's authority, the pope eventually assumed the highest level of authority, with bishops serving in their respective dioceses under his leadership.