An errand is a short trip to fulfill some small business. When you’re running errands, you’re making a bunch of these all at once.
The noun errand can refer to the mission of the trip itself. If the lines are always long at the bank, then banking may be your least favorite errand. If you are off on a wild goose chase, you may be out on a fool's errand. If you are taking food or clothing to someone in need, then you are on an errand of mercy.