I'm usually the first to apalogise when I've had a fight with someone and I just want to forget it ever happened. But the worst thing a bout apologies is when you know you're wrong, even if you have fought to the death about being right, sworn black and blue that the other person is in the wrong and blantantly lied about being right (because deep down inside you know you're wrong). Saying sorry after that is pretty embarassing and shameful; I would know, I have had to do this many many times.