When is it appropriate to apologize?

 I am probably the last person I am ever honest with. 

Does anyone else go over situations in their head to convince themselves that they are in the right? 

I do.

But then the line gets blurry for me. Because later I try to reconcile it within myself by understanding how I could have been wrong.

I try to understand how the other person could have felt like they were right, and how my actions could have affected their interpretations. 

And eventually after I practice this many times, in order to stay humble, I think I end up often apologizing for things that I don't need to be sorry about. 

How can you be fair to others and be fair to yourself? Especially in situations where it's clear the other person won't self analyze like you do.