I would say your mayor (in any city, town or village) influences your life more than any other politician-- along with your city/town/village council.
The decisions he/she makes can often be more directly felt in your everyday life than the decisions made by your congressman, governor, president or judges.
From property ordinances to millages to property taxes to economic development-- these are the things you deal with every day.
As a reporter and editor for the local paper for the past three years, I’ve seen first-hand what a power-hungry, egotistical, long-term mayor can do to a city (pop. ~98,000). Sure property values are continuing to rise, but many residents are getting pissed off at his micromanaging and the passing of dozens of strict property ordinances. He’s also taken an aggressive approach to economic development, buying a number of large vacancies around town and, with very little success, basically turned the city government into a real estate developer-- often wasting tax dollars, as well as keeping large properties vacant for years and years and years and years.
But as far as power goes, it depends also on the structure of your city’s government. Some cities have a strong-mayor system where the mayor is the administrative head, usually elected directly by the people. Other cities have a weak-mayor system where a council-appointed city manager runs the day-to-day stuff, and the mayor is a concilperson elected by the council to serve as mayor-- basically the council president.
But I think every city-head is equally powerful in relation to his or her constituents, some are just more prominent because they govern a city with more people, land or notable industries. Even if you aren’t paying attention to what your mayor is doing, chances are you’re feeling it and seeing it.