What is the difference between a polar covalent bond and a nonpolar one?

Polar character of covalent bond appears due to electronegative difference. Mostly covalent bond between unlike atoms give rise to polarity. Like atoms do not have electronegative difference hence they have non-polar covalent bonds.

A polar covalent bond is one in which the atoms are partially charged. As Chemist pointed out, this is due to electronegativities. As you move across the periodic table (and up it) towards fluorine, electronegativity increases. So water is an example of a molecule with polar covalent bonds between oxygen and hydrogen, as oxygen is more electronegative than hydrogen. These partial charges are usually shown with a lower case delta sign. (First link shows water with partial charges)

As such a non-polar covalent bond only truly occurs between atoms of the same element, eg the C-C bond in ethane.