Sumimasen (済みません): excuse me, I am sorry. It is the polite form of sumanai, which comes from sumu, to be finished or settled.
Mōshiwake nai (申し訳ない): I am very sorry. Mōshiwake means one’s case or explanation. With nai, it means there is nothing to say in one’s defense.
Owabi mōshiagemasu (お詫び申し上げます): I humbly offer my apology. The form used in formal written and public apologies.