The three forms of city government in Texas are the mayor-council, city commission, and council-manager systems. These typically differ as to who runs the executive branch of the city.
In the first form of government, the executive branch of the city is run by a mayor. There can be strong mayors who are elected separately by the people and are not part of the city council. They can typically do things like appointing various officials and vetoing laws passed by the council. There can also be weak mayors who are often part of the council and are only selected to be something like a chairman of that group.
In the city commission form of government, there is no separate executive branch. Instead, the various members of the city council each act as the head of one particular part of the executive branch.
In the council-manager system, the executive branch is run by an unelected official. This person is called the city manager and is hired for his or her competence rather than being elected by the people. This form is meant as a “good-government” form that is more technocratic.