There are two words loosely matching "errand".
* 심부름: an errand someone (parent, teacher, spouse, etc) has asked me to run. We say 심부름을 하다.
(But we never say 심부름 for our own business. It's always for someone else)
* 볼일: a thing to do (comes from "볼(to see) 일(matter)" - a matter to see about).
This is one's own business. We always say 볼일을 보다, not 볼일을 하다.
Examples:
- 나 엄마 심부름 하러 시내 나가: I'm going to the city to run an errand for mother.
- 선생님 심부름으로 왔는데요: I'm here to run an errand for my teacher.
- 볼일 보고 들어가느라 좀 늦겠다: I have an errand to run, so I'll be a little late.
- 난 볼일이 있어서 먼저 가야겠다: I'll have to take off first because I've got an errand to run.
- I had to run some errands for my dad today: 오늘 아빠 심부름을 좀 해야 했어.
* 볼일을 보다 is sometimes a euphemism for "relieve oneself", as in 그 친구 볼일 보러 갔어요 (He's gone on nature's call).