polite: 죄송해요 · past: 죄송했습니다 (no casual form)
I'm (very) sorry, I apologize
Example
오래 기다리게 해서 정말 죄송합니다.
I'm so sorry to have kept you waiting so long.
Word parts
죄 guilt+송 fear
Literally 'guilty and fearful' + 합니다 — more formal than 미안합니다.
Good to know
More humble than 미안합니다. Use it with teachers, bosses, elders and customers.
Related words