They're about taking responsibility for unintentionally (or even intentionally) hurting someone emotionally or physically. 
You apologize less because of you and your crime, but because of its effects on someone, usually someone you say you care about.

Apologies are not about right and wrong

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<img src="https://cdn.psychologytoday.com/sites/default/files/styles/article-inline-half/public/field_blog_entry_images/2018-05/images_2.jpeg?itok=juGEbTOn">We've all met people who never apologize, who proudly tell you that their family has a no-apology reputation that goes back generations. This tough stance is often mixed with notions of fierce independence, personal pride, and an every-man-for-himself mentality. Apologizing, according to this mindset, shows weakness and vulnerability; once you do, you're in danger of being taken advantage of.
For others, apologizing is less part of a <a href="https://www.psychologytoday.com/intl/basics/philosophy">philosophy</a> but more of a situation-specific determination of culpability that's clearly in the eyes of the beholder: "Did I really do something wrong? Even if you think so, I don't, so there's nothing in my mind to apologize for." Taking it one step further leads to a blame-game: "Whatever I did was only because of what you did-you're the one at fault here." This is particularly the psychology of those who are prone to <a href="https://www.psychologytoday.com/intl/basics/anger">anger</a> -"I only got angry because you..."
For yet another group, apologizing all too easily gets stuck in the mud of a power struggle, a fight-to-the-death standoff: "I won't apologize unless you apologize first. If you won't, I won't. Your call."
Even if your apology seems to put you in a weak position, the other person can't take advantage of you if you don't let them. And more importantly, apologies are not about right and wrong, an argument about which reality is right, but instead about something else: Taking responsibility for unintentionally (or yes, sometimes intentionally) hurting someone emotionally or physically. You apologize less because of you and your <a href="https://www.psychologytoday.com/intl/basics/law-and-crime">crime</a>, but because of its effects on someone, usually someone you say you care about.
Taking the step of apologizing is taking a step towards healing a wound, changing the climate of the relationship, getting the relationship back on track. And if you are a <a href="https://www.psychologytoday.com/intl/basics/parenting">parent</a>, it becomes a gift and a lesson that you give your children-about how to be responsible and compassionate in the world and towards others.
So how do you step up and be a compassionate and responsible apologizing adult? Here are some suggestions. 
Don't text Okay, you can text but the text shouldn't be any longer than, "I'm really sorry for..." One sentence. Any more and you run the risk of the other person reading emotions in your message that you don't intend and which ultimately kill your apology.
Send an email An email, the modern version of the written letter, gives you the emotional and visual space to talk more completely. You can take to time to think through what you want to say, and unlike a phone call, you're not suddenly putting the person on the spot and trying to have a conversation at a bad time.
A good email should cover three points: One is the apology itself, acknowledging that you are aware that you hurt the other's feelings and you're sorry for doing so. The second is providing some backstory, helping the other person understand what you were thinking and feeling at the time that prompted your words or your actions. This is important; in times of pain, the other person most wants to know what makes you tick, and without your explanation, the person is apt to make up their own incorrect one. That said, your explanation is not a green-light to be defensive and justify your actions. Keep your eyes on the prize, namely, wanting to heal a wound. If you find yourself drifting into defensiveness, you're not ready to write the email.
Finally, you want to include in your email whatever you think the other person might begin to think. Here you say things like, I'm not trying to justify my actions, I'm not saying that this was your fault, etc. What you are doing by including these statements is clarifying and hopefully calming the other person's possible misreadings and reactions in advance.
Follow-up Don't just send the text or the email and think you're done. They lay the foundation for a follow-up conversation. The purpose of a face-to-face or phone conversation is to hear the other's reaction, to clarify what was misinterpreted or not understood, and to gauge whether what you hoped to do was actually done.
What if the other person doesn't accept your apology? Circle back. Maybe there is something, despite your good efforts, that they didn't understand or misinterpreted. Resist the urge to push your point or get defensive and huffy in response. At a certain point, realize that you have done the best you could. Time to move on.
Apologies are not about graveling or beating yourself up, about losing face or admitting defeat, or a manipulative tool for getting the other to admit fault. They're about repairing hurts and wounds and <a href="https://www.psychologytoday.com/intl/basics/relationships">relationships</a>. 
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We've all met people who never apologize, who proudly tell you that their family has a no-apology reputation that goes back generations. This tough stance is often mixed with notions of fierce independence, personal pride, and an every-man-for-himself mentality. Apologizing, according to this mindset, shows weakness and vulnerability; once you do, you're in danger of being taken advantage of.

The Art of the Apology

Apologies are not about right and wrong

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