Things to do and not do when apologizing

Avoid:
-using the word "but" as it can invalidate a sincere apology. (I'm sorry that I...but")
-using the word "if" as it can suggest that a hurt never happened. (I'm sorry if you felt hurt...)
-justifications and explanations as it can muddy up an apology and cause confusion.
-generalizing with vague wording as it can sidestep taking responsibility.

Instead:
-first listen to the person's hurt and understand it from their perspective.  Repeating is a great way to communicate that you understand and can also clear up any misunderstandings. (I heard you say....)
-own up to your wrong doing and assume responsibility.
-be specific in your apology for the action and words that offended the other person.
-make proactive attempts to rectify a conflict if you are in a position to do so by "making up" for the wrong doing.