How can you determine if the bond between two atoms is polar covalent, nonpolar covalent, or ionic?

The textbook answer is to consider the difference in the Pauling electronegatitivies of the two bonded atoms.  If the difference is less than about 0.4, it's essentially nonpolar.  If it's between 0.4 and about 1.7, it's polar covalent, and if it's greater than 1.7 its mostly ionic.  

weak bonds are important in biology because there are a lot of them.  In a protein for example the overall structure is largely determined by a very large number of relatively weak bonds such as hydrogen bonds.

Polar Covalent bond: Valence electron is shared between two atoms. But because one molecule is slightly more electronegative than the other,electrons spend more time around the more electronegative side. Results in slightly opposite charges on the molecule's ends. The electrons are not shared equally.
Example: H20. Hydrogen side is slightly more positive than the oxygen side. Because oxygen is more electronegative the electrons spend more time around it, thus causes a slight disparity in charge.

Nonpolar covalent bond: Valence electron shared between two similar atoms. Because the two atoms have similar properties, the electrons are shared equally between them.  
EX: H2, O2.

Ionic Bonds: Interaction between oppositely charged ions [cations and anions]. Ionic bonds form salts.

-Covalent bonding leads to a definite size and number of atoms in the formed molecule.
-Ionic compounds only have ratios - ex: NaCl [1Na : 1Cl]

Hydrogen bonds are an example of a weak bonds.
They may be weak, but the unique properties of water are all due to waters many hydrogen bonds.