A mayor's job description depends on the area they preside over and whether their role ties with the government or the local council. Those working with the local council only may fulfill a part-time role that allows them to represent the people of their town in local matters, whereas those with a strong governmental role preside over city boards and commissions.
Mayors working in a small capacity are the official head of the city, but have a part-time role. This usually happens in small to medium size cities. Such mayors play a role in chairing the city council, but do not have legislative powers. Their activities include controlling the city's liquor, appointing treasurers and overseeing small council matters.
In contrast, mayors of larger cities, like New York City, play an influential role in government. They may oversee a team of commissioners and city boards. They are also involved in the city's legislative process, and can veto laws the council passes. The council can subsequently override this with a two-third majority vote. The mayor also appoints judges to family, civil and criminal courts. They may also sit ex-officio on certain boards.
Most mayors in America, whether large or small, are referred to as "His honor" or "Her honor."