To make a meaningful apology one has to make sure to include all the parts.....the anatomy.
The first part of the apology is where you state that you are sorry and adjunct to that, exactly what it is you are sorry for. For example:
I am sorry for breaking your vase
Clear and to the point. But that is really not enough. You also have to acknowledge that you understand that it was wrong, inappropriate, hurtful, or whatever.
It was very irresponsible of me
Next you need to offer a way to make it better. A compensation. This will not fix the wrong, but does at least show that you do care about the other person's feelings.
Let me try to replace it or fix it for you
And finally, you need to finish up with a statement of intent to not repeat the offense. This is the way to show that you truly understand the impact of what you did and that you have enough regard and respect for the person to not want to ever cause that discomfort again.
I will be much more careful in the future
When all these parts are put together with sincerity and eloquence, the apology is much more heartfelt and it is received much more readily. It is those things that make it an art.