Saying sorry is incredibly difficult for some people - and is something that sometimes needs to be learnt. Once you have mastered the art of apologizing (well lets hope you don’t need to do it too often! lol) it makes you a much more desirable person.
The three key elements to a true apology
1) Sincerity - a sincere apology is true, genuine and pure. Before you apologize you need to understand what it was that you did wrong and how you hurt the other person - and most importantly that you are truly sorry for what you did. An insincere apology can be really offensive and will show when you make your apology.
2) Take responsibility for your actions - Don’t make excuses. Take full responsibility for your actions. When you add a justification or an excuse to your apology you strip it of sincerity.
3) Promise that you will try to do better - When you apologize it means that you don't ever want to cause that person pain again - therefore when making an apology you need to attach a promise to it - that promise is not a rash one - i.e. That you will never do it again but a promise to try harder to make sure that it never happens again. Keep that promise and work on your weaknesses so that you fulfill it. This is also an important act in re-establishing trust.
There’s another benefit for you when you learn the art of sincere apology - people respect you. It takes courage to admit you’re wrong.