how to apologize

What is it with apologies? Why do some feel hollow and empty while others elicit understanding or even forgiveness? What are the specific qualities of an authentic apology?
Consider this. We all make mistakes, but there's no reason for your apology to be a mistake of its own.
Is it too late now for Donald Trump to say sorry? According to Canada's rules of politeness, maybe not.
"I never meant to hurt you."
I didn't realize that the habit of starting sentences with, "Sorry, but could you..." actually made me present my own needs as weaknesses. It was probably clear to the other person, but the reason it wasn't clear to me was because the word "sorry" had subconsciously made me feel as if I was constantly bothering other people.
Presidential candidate Jeb Bush has confessed, given what we know now, he would not have authorized the invasion of Iraq, as his brother did. A politician with integrity should have followed that comment with an apology to the Iraqi people.
What happened to saying no and not following up with an explanation? What happened to just saying, "I don't feel like it"?
As Paula Spencer Scott, wrote in 7 Ways to Handle a Hothead -- Without Blowing Your Top steadying yourself with people who feel slighted is challenging. In my line of work, this would be hurt and angry co-parents and their children.
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I never learned to say "I'm sorry." In my household growing up, it wasn't said too often. But learning to say it, and say