Mayoralities are based on the concept from FourSquare.
If you are the cacher with the most GCA finds for a locale/state/country combination and that count is >= 10, then you are declared the mayor of that locale. If there is a draw for the same number of finds, the one who has the latest find gets to be mayor. i.e. The newcomer usurps the old guy.
Locationless caches and moveable caches do not contribute to the mayoral status as locationless don't have a location to be mayor of and moveable caches, well, move.
Neribil has 10 GCA finds in Ulverstone and as the highest finder, gets to be the mayor.
You can see the current mayor next to the Mayoral Crown.
He (she, it) is also the mayor of a few other locales.
You can also see that Neribil is a mayor on the "bling line" on their cacher page.
You can see whether you are a mayor by going to your cacher page and the under Statistics you can select Mayoralties.
Are you a mayor yet?
Can you usurp the incumbent and take over the mayoral role for a locale?
This section lists the mayoralties that the cacher has the potential to take over. The mayoralties may be unoccupied or there may be a mayoral occupant in place. Mayoral potential is calculated on a minimum number of 5 finds. Any locale where you have less than 5 finds will not be shown on your potential list.
If the cacher has 10 or more finds in that mayoralty, then they have reached the minimum number of finds to become mayor, but there is a mayor already in place. More finds will be needed to usurp the incumbent mayor. A palace revolt may be needed to usurp the incumbent.