In professional wrestling, a run-in occurs when one or more individuals who are not actively participating in a match run into the ring. The vast majority of run-ins are made by heels, typically to further a feud with a face.
More often than not, a run-in will result in a "beatdown" in which the heel(s) pummel the face(s) until the script calls for the beating to stop, either from the heels' satisfaction with their handiwork, a retaliatory run-in by one or more faces, or (less often) the entrance of one or more authority figures (referees, road agents, security personnel, et al.).
Sometimes a run-in results from a face wanting to stop a heel from physically punishing a weaker opponent, usually to set up a feud. Run-ins have also been known to involve authority figures doing the same. A good example of this was seeing Jerry Lynn "run-in" on Kevin Nash in Total Non-Stop Action, this was used to help create "heat" on Kevin Nash.
However, not all run-ins result in beatdowns. One especially famous example of a run-in that did not lead to a "fight" was the MSG Incident