What happens to an electron's potential energy when it moves away from the nucleus? How about when it moves toward the nucleus? What is the biological significance of this situation?

An electron has the more potential energy the further it gets from the nucleus. Think of it like stairs: A ball on top has the potential to move down lets say 10 stairs. That's a lot of potential. If it drops down a stair, it can only go nine more. Each time electrons move down an orbital, they release energy (exergonic). It is easier to take an electron from the outer rings (valence electrons) than those closer to the nucleus, because they are held tightly and are very stable. It would require energy to theoretically take those electrons farther away. Atoms want to fill all of their rings to the max to become stable, so they need the electrons to have the least amount of energy (that would enable them to leave the atom).