In school, after finishing a course you receive a grade (A, B, C, D, F=fail). The grades A, B, C, D are "passing grades". "Giving someone a pass" means giving them a passing grade, even if they didn't do a great job and maybe they really deserve a failing grade. More generally, the expression is used to mean overlooking poor performance.

Jonathan gives a good explanation of this phrase. I would just add that it can also mean to overlook bad behavior. It is similar to the word "forgive." 1. My husband forgot my birthday, but I gave him a pass because he has been working a lot of hours lately. 2. I was late for work today, but my boss gave me a pass because I've never been late before.