A mistake is an error, a goof, a slip-up. When you make a mistake, you've done something incorrectly.
Mistake has a lot of uses, but they all have to do with doing the wrong thing. A mistake in math class will result in the wrong answer, but a mistake with a gun could get someone killed. If you mix up two people, you're mistaking one for the other. When you make a mistake, the best thing to do is admit it and try to make up for it; not apologizing for a mistake is another mistake.