The title held by the leader of Michel Delving, the 'capital' of the Shire. Over time, the power of the Mayor grew to encompass all the lands of the Shire.
We have no way of knowing when Will Whitfoot's time as Mayor began, but the latest this can have been was in III 3013, the date of the last Free Fair before the War of the Ring, and thus the last point he could have been elected. As a popular and distinguished old Hobbit, though, it's implied that Old Will had held the position for rather longer than the fourteen years we can establish for sure.
2
Tolman Cotton (eldest son of Farmer Cotton) had served as Sam's Deputy at one time, so he might be considered a plausible candidate as Sam's successor, if not for the fact that both these Hobbits had been born in the same year (III 2980). At the age of ninety-six, then, Tolman would have been a little past his prime, and it seems more likely that a member of the next generation of Shire-hobbits would have been elected to the Mayoralty.