A run-in is a confrontation or an argument. Peaceful protests go more smoothly when protestors avoid run-ins with the police.
If you get in a fight with your brother about how he never loads the dishwasher, you can call that a run-in. A more serious kind of run-in might be a shoplifter's run-in with a store security guard or a basketball player's angry run-in with a referee that gets him benched for the rest of the season. Before 1905, run-in meant an instance of actually running somewhere.