Berkeley has a council/manager form of government, as opposed to a “strong mayor” government as bigger cities like Oakland and San Francisco have.
This means that power is divided between the city council, which includes the mayor, and the city manager, who directs the city staff. The council sets city policy, and the manager administers the government according to the law and council policy. The roles of the council and manager are defined in the city charter, which is the constitution of the city.
We also use the term city administration to refer to the city manager and all the city management staff, including the police department leadership.
The city manager is not elected, but is appointed by the city council. But she is effectively the most powerful person in city government. Even the mayor has only one vote on the city council. The only accountability the manager has, as to whether she is actually fulfilling the council’s policies, is to the city council, which has the power to remove her by majority vote.