Axiom of choice question
I know there is a lot of discussion on the axiom of choice and, in fact, I attended once a lecture on it, but I still cannot understand the following: Let $A$ be a nonempty set. I want to pick an element from $A$. Since by hypothesis $A$ is nonempty, it must contain at least one element, $a$. So I pick this $a$. Where is the "gap" or the need for the axiom of choice?