It depends on how formal you want the apology to be and who it is directed at, really. For example:
You can say:
실례합니다 shil-lye-ham-ni-da which is a very formal way of saying it, perhaps to an older Korean person, or when you want to be respectful to someone
미안합니다 Mi-an-ham-ni-da but you would only use that one to a friend, an older person would be insulted.
Interestingly, you would not say 'I'm sorry' to a Korean who had just told you they had some bad news, they would wonder how it was your fault and why you were apologising. A lot gets lost in translation doesn't it? But I find learning new languages fascinating!
I have included a video so that you can hear how the phrase sounds and understand the different ways of expressing yourself in Korean.
