When you apologize you admit that you did something wrong, and you say you’re sorry. The point of apologizing is usually to clear the slate and explain why you did what you did.
We all make mistakes. Sometimes we even do bad things on purpose. But it can all be okay, so long as we apologize. Apologize comes from the Greek apologizesthai, which might be impossible to pronounce, but which means, “give an account.” When you accidentally run over your neighbor's flowerbed on your bike, you better be ready to apologize.