One pays a fee as renumeration for services, especially the honorarium paid to a doctor, lawyer or member of a learned profession. As part of their mystique, traditional professionals in Britain received a fee in contradistinction to a payment, salary, wage or mere money, and would often use guineas rather than pounds as units of account.
One may also charge /pay fees as a fixed sum for the right to enter for an examination, or on admission to membership of a university or other society.
For technical (and original) usage of the term fee in the European feudal system see fiefdom.