Elementary

1. Sorry.
2. I'm (so / very / terribly) sorry.
3. Ever so sorry.
4. How stupid / careless / thoughtless of me.
5. Pardon (me)
6. That's my fault.
7. Sorry. It was all my fault.
8. Please excuse my (ignorance)
9. Please don't be mad at me.
10. Please accept our (sincerest) apologies.
11. Phrase 1 is a general short apology. We use this when we bump into people on the street. At other times, it sounds too weak.
12. In phrase 2, we use 'so', 'very' and 'terribly' to make the meaning stronger. 'Terribly' is the strongest. If we use one of the words in brackets, it is stressed.
13. Phrase 3 is quite formal but it's a stronger apology than just 'sorry'.
14. We use phrase 4 to criticise ourselves and the mistake that we have just made.
15. We use phrases 6 and 7 to take all the responsibility for what happened. Phrase 7 is a little stronger.
16. We use phrase 8 to apologise for our lack of knowledge or ability. We can replace the word in brackets with other nouns, e.g. carelessness, forgetfulness.
17. Phrase 9 is asking the other person not to get angry. The tone is quite informal.
18. Phrase 10 is often used in formal letters. The word 'sincerest' makes the apology very strong and very formal.