Japanese etiquette holds that one should apologize by saying the equivalent of “I’m very sorry.” But a proper American apology requires something more: a statement of exactly what one is sorry for and often an explanation for why one acted that way.
For example, if you are late for an appointment, you could say, “I’m sorry to have kept you waiting. I just missed the bus.” If you have to break a date, you could say, “Jane, I’m sorry I can’t come to your party like I said I would. An important client just called and said he would like to meet with me that evening. I’m awfully sorry to back out at this late moment.”
If you hurt a person’s feeling, you can say, “I’m very sorry that I hurt your feelings the other day. I wasn’t thinking and I want to apologize.” A sincere apology with an appropriate facial expression, not a smile, is usually enough to repair the damage. Merely repeating “Sorry. Sorry.” will not get the job done.