1. Literally, to run haphazardly or in different directions in an area. Primarily heard in UK. We took the kids to the park so they could run about for a couple of hours.
2. Figuratively, to be exceptionally busy doing many different tasks or activities, especially in several different locations. Primarily heard in UK. I've been running about all afternoon getting things for Carol's party this evening. I can't wait for the weekend. I've had to run about all week getting little jobs done for my boss, and I'm exhausted!
See also: run