What is the meaning of "run someone out of town"?

The situation which comes to mind is the Sheriff in a town in the Old West (of the US) who has to get some bad guys to leave town - he has to run them out of town.

He might put them on a train, or put them in a stagecoach, or simply tell them to get on their horses and ride.

It's a little unusual for a woman to run the mayor out of town, but perhaps he was corrupt, and the ladies of the town shamed him into leaving.

Most obvious meaning is that she caused the mayor to have to leave said town by or as if by force; either physical, or perhaps by sullying his reputation to the extent that he feels he has to leave, as he's been given such a bad name.

chase out of town, also bum's rush

The expression does have (U.S.) "Wild West" connotations, and it basically means to "eject" someone from town.

This usually occurred when someone was carried out of town "on a rail" by a mob, often after being "tarred and feathered" (i.e. covered by the same). This process was so forceful, and attended by so many people, that only the most intrepid or foolhardy would ever set foot in town again.

The context appears to be a modern one, in which a woman probably got together a "gang" of citizens (in e.g., a "recall" motion), or possibly a "posse" of lawmen, (in e.g. a corruption case), that had the effect of not only ejecting the mayor from his office, but essentially forcing him to leave town altogether.

I assume neither OP nor most readers have any problem with the 'literal' meaning of the expression (she and/or her supporters forced the mayor outside the town limits, with instructions never to return).

In practice I think that's not a plausible scenario today. The usage is clearly metaphorical, indicating merely that the mayor was roundly defeated on some issue. Most likely an election for a new mayor, wherein the woman in question won convincingly. It's not likely she used her new powers to banish him from the town; I doubt US law would allow such petty vindictiveness (though I'm sure knowledgable commenters will wish to contest that point).

In modern context

To harass someone so much that they are forced to leave the area to escape harrassment.

This is generally a response to unacceptable behavior.

It is also used in context of ending or preemting challenges to political or social standing.

This is actually a shortened version of "run out of town on a rail.". This is sort of a public humiliation punishment used in the USA many years ago on people who perhaps had committed no actual crime worth imprisonment, but had somehow made their continued presence in town undesirable.

The offender would be exiled from town by forcing them to straddle a fence rail (or later a railroad tie) while men hoisted it on their shoulder and paraded the offender through town to the town limits, so that the whole town can't help but see what is going on.

There's a good scene of this happening near the end of O Brother Where Art Thou?

When used without the rail, it implies that the person in question was either physically or emotionally coerced into leaving. I sometimes hear it used when there is no actual town involved (eg: When someone is harassed into quitting a job)