1. To have incriminating or unfavorable proof against someone, as for some crime, wrongdoing, or misbehavior. Go to the police, for all I care! You haven't got anything on me! If we want to nail him for fraud, we'll need to get more on him than a few questionable phone calls.

2. To have a particular advantage over someone. You may think you can beat me, but I've got 10 years' experience on you.