The responsibilities of the mayor are likely to vary significantly from city to city. Some cities are likely to have a weak mayor and mainly accepts the responsibilities of presiding over ceremonial duties, whereas you have a strong mayor with wide-ranging responsibilities.
In essence, the two main types of mayor sitting in office are likely to relate to the mayor-council and council-manger, with the distinction between these two forms relating to the degree of authority and influence in the city that the mayor holds.
Mayor-council: A mayor in the position of mayor-council is the stronger of the two forms and sits in the largest towns and cities across the US. In this role, the mayor takes on the responsibility of the manager of the city or chief executive. They aren’t likely to form part of the city council and therefore unable to influence lawmaking policies. But, this type of mayor does still have a great degree of self-rule and administrative power in a lot of situations.
Council-manager: If residing in one of the smaller towns or cities, the mayor might take on the role of a council-manger mayor, which generally involves responsibilities relating to ceremonial activities. A mayor in his position is unlikely to have any more weight or influence in the decision-making responsibilities for the city, then any other member of the council, although they do take on the role of presiding over the meetings.
In relation to the general duties taken on by the mayor, these might relate to the ability to propose the budget for certain city expenses, the ability to hire of fire a city employee, monitoring local services, such as those relating to the roadways and parks, and maintaining and creating positive business relationships in the local community. The listed duties for the mayor are likely to be quite similar in most locations across the United States.