The axiom of choice says that for an infinite set of disjoint nonempty sets, we can find a set with one element from each. In others we can choose an element.
The axiom of choice is needed only when there is no clear way to choose. For example if the disjoint sets were integers, we could always explicitly pick the smallest, but for sets of sets, there isn't a clear way to pick.
Axiom of Choice = Cartesian Product
It postulates the existence of a set which is not uniquely determined, of which we know only some properties.