There are typically said to be three forms of government that are used by cities in Texas.
The first form is a mayor-council system of government. In this system the city council is the legislative branch while the mayor runs the executive branch. All of these are elected offices. Some texts differentiate between strong mayor systems and weak mayor systems. Most cities in Texas have weak mayor systems.
The second form is a city commission form of government. In this form, there is no one leader and each department of city government is run by an elected city commissioner. There is no separate legislative branch.
Finally, there is the council-manager system. In this system, there is an elected city council that is the legislative branch, but the executive branch is run not by an elected mayor but by a professional city manager hired by the council.