Cities in the United States are sometimes categorized as having either "weak" and "strong" mayors. Most "strong" mayors are in the mayor-council form of government, and are directly elected by citizens to that office. Most "weak" mayors are mayors in a council-manager form, and are elected by and from the council.
The distinction is in the level of political power and administrative authority provided by the city charter to the mayor. In actual practice, there is no sharp line between "weak" and "strong" mayors. Rather there are continua of authority and power, along which cities are spread.
Yet, it is useful to contrast the differences.
Characteristics of a "weak" mayor:
The council is powerful, with both legislative and executive authority
The mayor is not truly the chief executive, with limited power and/or no veto power
The council can prevent the mayor from effectively supervising city administration
There may be many administrative boards and commissions that operate independently from the city government
Characteristics of a "strong" mayor:
The mayor is the chief executive officer, centralizing executive power
The mayor directs the administrative structure, appointing and removing of department heads
While the council has legislative power, the mayor has veto power
The council is relieved of day-to-day administration
Sources:
DeSantis,Victor S. and Tari Renner. "City Government Structures: an Attempt at Classification" State and Local Government Review, vol. 34, no. 2, (Spring 2002).
Forms and Structure of Municipal Government in the United States, presentation by William R. Barnes to the Convention of Scottish Local Authorities on October 24, 1991, in Edinburgh, Scotland.
Frederickson, H. George, Curtis Wood, and Brett Logan. "How American City Governments Have Changed: The Evolution of the Model City Charter" National Civic Review. vol. 90, no. 1, (2001).
Frederickson, H. George, and Gary Alan Johnson. "The Adapted American City: A Study in Institutional Dynamics." Urban Affairs Review vol. 36, no. 6 (July 2001).
National League of Cities. Choices of the Citizenry: Forms of Municipal Government. Washington, DC: National League of Cities, May 1989.
Svara, James H. "The Shifting Boundary Between Elected Officials and City Managers in Large Council-Manager Cities" Public Administration Review, vol. 59, no. 1 (January-February 1999).
Svara, James H. Two Decades of Continuity and Change in American City Councils. commissioned by the National League of Cities, September, 2003.