Definition
To force someone out of a place or a position.
How People Actually Use It
Olde-timey phrase, most commonly referring to the act of literally carrying someone perched uncomfortably on a rail to a point outside of the city limits. This was often a form of punishment for committing any act others thought was extremely bad. It is still sometimes used in the Southern states.
"Doctor:What did she take? Friend:These, Dexamil. Supposed to cure drinkin' Doctor: Alan T. Whitman, M.D. That man ought to be run out of town on a rail."
Source: Urban Dictionary