Apologizing doesn't always mean that you're wrong and the other person is right. It just means that you value your relationship more than your ego. 
We must remember an apology isn't an apology unless it's meaningful! If your apology isn't sincere and you're just apologizing to avoid an argument then that can just lead to more damage. Also you can't just apologize for everything, you have to know what you're apologizing for. You have to be able to admit you're in the wrong and mean it. Apologizing can be very relieving for all parties involved including yourself.