When I have been wronged, insulted or the victim of larceny I am usually quite vocal and ask for the situation to be corrected or at minimum recognized.
Of course the perceived wrong doer prefers to do nothing but often offers the worst of all apologies...
"I'M sORRY YOU fEEL tHAT wAY!"
That is worse than no apology. Its actually just another insult for you being so foolish as to ask for an apology and ignores the issue that is in question.
Well leave it to the Pope to offer the "I'm sorry YOU feel that way" apology, which has led to 5 church burnings and a 17 year old pediatric care Catholic nun by Islamic extremeists.
How can one best apologize? First review the dispute with the offended party and come to a bilateral and complete understanding of the issue immediately and ask exactly what words or action would best resolve the dispute. Follow up and do the right thing.
Now as for the ass holes who did the exact things the Pope was talking about when he referred to writing over 1,000 years old about the violence of spreading Islamic beliefs among unbelieving infidels...