Reconciling After A Fight: How Do You Say "I'm Sorry"?

Relationships that could be saved often break up not because the people no longer love each other, but because neither party knows how to apologize after an argument. If you grew up in a household where people bickered and argued and even physically fought, but nobody ever demonstrated how to reconcile after the conflict, how would you know the process? Someone has to choose to be the bigger person and step forward to heal the hurt, whether or not they were the one most in the wrong.  When neither party will do so, whether out of anger, unforgiveness, or fear of rejection, things remain at a stalemate.  In a marriage, the stalemate will eventually result in divorce.  And yet, most marriages could probably be saved if people were able to talk to each other and resolve whatever caused the hurt. How do you reconcile after a fight? How do you repair a broken relationship? Are some relationships shattered beyond repair? If you are in conflict with someone, are both of you waiting for the other one to apologize first? Are you willing to be the bigger person and forgive? How do you say "I'm sorry"?