1.
used when you are apologizing for sth :
I'm sorry, I forgot.
Oh, I'm sorry. Have I taken the one you wanted?
I'm sorry. I can't make it tomorrow.
2.
used for disagreeing with sb or politely saying 'no' :
I'm sorry, I don't agree.
I'm sorry, I'd rather you didn't go.
3.
used for introducing bad news :
I'm sorry to have to tell you you've failed.