Sometimes we forget really easy lessons. Whether you're familiar with Daniel Tiger or his real life predecessor - Mr. Rogers - the lessons we learned in pre-school (or before school for those old enough to remember a world before pre-school) still hold true as you grow older.
Saying I'm sorry is the first step. It's a step that is extremely difficult for many people. It is also one that comes too easy for other people - those looking to avoid conflict.
How can you have the correct level of sorry where it is sincere and meaningful?
The answer is simple, add the second part of the Tiger's example - "And how can I help." If you are not willing to do this second part, then your apology was more an apology of convenience.
If you're a person who can not muster the first part, then you most likely owe quite a few people an apology. Try it, I guarantee you'll feel better and your friend approval ratings will soar.
You're welcome. - James ;)