1. To travel across to where someone or something is very quickly, as by running. Will you run over to Jerry and give him this note? I need to run over to the post office to send this package.

2. To deliver or convey someone or something to someone, something, or some place. In this usage, a noun or pronoun is used between "run" and "over." I need you to run these documents over to the courtroom as soon as possible. I'm just running Timmy over to soccer practice, so I'll pick up milk when I'm on my way home.