Quality apologies are rare, and even rarer when it comes to adults apologising to children. Some of us might be afraid we will be seen as weak, and not worthy of being taken seriously in future.
  But when someone feels wronged or disadvantaged by inappropriate behaviour they need to feel safe in the knowledge that the incident won’t happen again. They require an apology: one that indicates genuine remorse and change in the person’s behaviour.
  Though let’s be clear: I’m not suggesting we apologise so that the other person benefits, I’m suggesting it because we benefit. This isn’t a book on ethics, it’s a book on how to build a person and gain resilience.
  By taking full responsibility for our actions, by fully acknowledging that what we did was wrong, we significantly increase the likelihood that we won’t make the same mistake again. (Shame has a lot going for it!) As a consequence, we become more competent and more self-assured, and bolder. We start to make better decisions. We earn back the trust and respect of others, too, because they also know we are less likely to make the same mistake again. That strengthens the connections we have with people.
  Further, by acknowledging our faults – to ourselves and to others – we can lance the ‘pus ball of shame’ within us and experience relief. Shame has done its job. It can now return to the Dark Forest.
  And, having allowed ourselves to experience that vulnerability, we remind ourselves that we can cope with the consequences, so we add to that feeling that whatever happens, we’ll handle it.
  That is why having the ability to give a full, considered apology (only when it is due) is a big confidence booster, and a key to resilience and core happiness.

Q. ‘How can I give someone a quality apology?’
Try here: ‘How to give a quality apology‘
and here: ‘How to give a lousy apology‘.

Q. ‘Some people apologise all the time. It ends up not meaning anything.’
Good point. Don’t over-apologise. Especially avoid the habitual ‘sorry’ some people murmur often.

Q. ‘I gave a written apology to my school teacher. How will I know if she accepted my apology?’
  You may not come to know. That’s none of your business.
  ‘Why not? I wrote the letter!’
  The recipient of your apology owes you nothing. Your teacher will decide whether or not she has accepted your apology, and whether or not she has forgiven you, and whether or not she tells you of her decision.
  The purpose of your apology is to take responsibility for what you have done, and to help your teacher feel better. Its purpose is not to earn her forgiveness. It’s not about you.
  ‘I don’t benefit?!’
  You do. Each time you take responsibility for what you have done you confront your own fallibility, and are reminded that you are not diminished by it. And, coming to terms with your own fallibility will mature you and grow your confidence in a healthy way. With your more cautious self-assuredness you will  make better decisions in life.

Q. ‘What if I don’t think I should apologise, but it is expected?’
Go over the incident thoroughly.
(1) Aim to see the incident from the other person’s perspective. What would they be thinking? What would they be feeling? How hurt might they be?
(2) What would you think and how would you feel if that were done to you?
  Then, if you still believe you should not apologise, don’t. If you like, politely explain your reasons.

Q. ‘What if the person is still not happy after receiving my apology?’
If your apology is a quality one, and you have done your genuine best to make amends within reason, there is nothing more you can do.

Q. ‘What if the other person is partly at fault?’
The part the other person played is irrelevant to your apology. Apologise for the part you played in the incident. Don’t qualify it by mentioning the part they played. Make that a separate discussion. An apology needs to be uncontaminated.

Q. ‘When can I demand an apology?’
Before you demand an apology ask yourself why you want it. Is it to exercise power over the person because you are in the right?
  ‘I want to make a stand. I want the person to think twice about doing it again.’
   You can demand an apology but you may not get one. That could make you feel worse. Don’t ask for one unless you are sure you can handle not getting one.
  If you don’t get one, don’t dwell on the fact.
  ‘I suppose if I do demand one and get it, it will be lousy and full of resentment.’
  Not always. But remember, most people aren’t practised in giving apologies. Usually they try to justify their position, so there is a good chance you will feel worse afterwards.
  A better idea is to make sure the person knows they have wronged you. ‘You are unacceptably late and I am inconvenienced‘, or ‘Your remark was cruel.‘ If you can state your case clearly and comfortably, you are sticking up for yourself and that will help you feel better. It can be a better reward than any apology you might receive.

Q. ‘If I throw a cricket ball at the stumps and accidentally break someone’s toe, do I have to give a full apology and intend to change my ways?’
We don’t need to make a full apology when we unintentionally hurt someone, unless it’s by negligence. When we say sorry in such an instance we are not apologising for what we did, we are expressing concern.
  ‘So, Jill tells me her father just died. If I killed him I give a full apology. If I didn’t kill him I can say sorry to express my concern that she’s suffering.’
  Er, yes.