If you do or say something that hurts or offends others, it is polite to offer apologies. In English we use several different expressions to offer apologies. The most common among them are given below. 

I’m really sorry…

I’m sorry…

I’m extremely sorry…

I’m so sorry…

I’d like to apologize…

I know I shouldn’t have done this to you.

I do apologize for…

I apologize for…

I’d like to apologize for…

I must apologize for…

I shouldn’t have…

Examples are given below.

I’m so sorry. I didn’t realize that I was stepping on your foot.

I’m sorry for disturbing you.

I’m sorry for breaking the glass. I couldn’t help it.

I’m so sorry. I shouldn’t have said that.

I’ve realized my mistake. Please accept my apologies.

I’m sorry. I didn’t realize this was your seat.

In a more formal style you can say:

I beg your pardon

I can’t say how sorry I am.

Pardon me!

Please forgive me.

I’m sorry to have disappointed you.

Please excuse my thoughtlessness.

I understand that it was my mistake. Please forgive me.

There is really no excuse for my actions / behavior. I just hope that you’ll forgive me.