The main form of organization of medieval society was known as "feudalism." Within this system, people were divided into three "estates", the nobility, the clergy, and commoners.
Society in this period was extremely hierarchical. Among the nobility, the king was preeminent, in theory owning all of the land in a kingdom, and giving use of it to nobles who were his "vassals" in exchange for rents in the form of either goods, money, or services (especially military service). In some countries, the power of the king was absolute but in other nobles and kings vied for power. Sometimes the king would ally with rich merchants to thwart the nobles. In the cities, there were artisans and merchants who, although they were "commoners", might amass wealth and power. Many cities operated under charters which gave them certain levels of independence. In the countryside, most people who were not nobles were peasants or serfs, tied to the land, and working for the noble who held the land.
In terms of religion, Europe was Roman Catholic. The Pope was the head of the Church, and Bishops were the equivalent of clerical nobility. All of Europe was divided into dioceses supervised by Bishops and subdivided into parishes each with a priest. The Church was responsible for baptism, marriage, and burial, and maintaining parish records and taking care of the poor in the parish. Education was normally either via church-run schools or apprenticeships to master craftsmen. There was very little secular bureaucracy or government in the modern sense, as many of the tasks such as registering birth were handled by the Church.