If somebody is running an errand, they are making a short or medium distance trip, or trips, to accomplish some purpose - usually to get a thing or number of things and bring them back. "She's away right now running errands. She's going grocery shopping and dropping off the laundry." "I have to run some errands. We're out of coffee and I have to pick up some light bulbs." "I have to run a quick errand. We're out of milk."

"to run errands" = to do small jobs which you have go buy or deliver something. i.e go to the grocery store to buy groceries, go to the post office to post some letters, go to the bank to make some deposits.