Electronegativity determines how much an atom wants electrons. The more electronegative an atom, the more it wants electrons. This is important to keep in mind when looking at the different kinds of bonds.

If one atom is much more electronegative than another, then it can either completely take an electron from the other atom (ionic bond), or it can simply pull the electrons toward itself more (polar covalent bond). As a result, covalent bonds that contain atoms with very high electronegativities (like oxygen or fluorine) are polar. The oxygen or fluorine hog the electrons.

This is the basis for the difference between polar and nonpolar bonds. The unequal sharing of electrons results in the bond having a partially positive end and a partially negative end. The more electronegative atom is partially negative (denoted δ^-) while the other end is partially positive (denoted δ⁺).

Bonds can either be completely nonpolar or completely polar. A completely polar bond occurs when one of the atoms is so electronegative that it takes an electron from the other atom (this is called an ionic bond).

On the other hand, when the electronegativities are exactly the same the bond is considered to be a nonpolar covalent bond. The two atoms completely share electrons.

But what happens in between these two extremes?

Here is a table that demonstrates what kind of bond is likely forming based on the difference in electronegativity:

Thus, the difference between polar and nonpolar bonds is due to the electronegativity difference of the atoms.

A compound can have polar covalent bonds and yet still not be a polar compound. Why is that?

Polar compounds have a net dipole as a result of polar bonds that are arranged asymmetrically. This means that they have both a partial positive and partial positive charge that do not cancel out. An example of this is water.

Nonpolar compounds can either entirely share their electrons, or they can have symmetrical polar bonds that end up canceling out any sort of net dipole. An example of this is BF₃. Because the polar bonds are arranged in a single plane they end up canceling out.

Chemical polarity plays a huge role in how different molecules interact. For example, why does sugar dissolve in water while oil does not?

It's all about polar vs. nonpolar.

Water is a polar solvent. The oxygen atom contains two lone pairs and is more electronegative than hydrogen, thus pulling the electrons towards itself. As a result the oxygen atom has a partial negative charge associated with it. The hydrogens on the other hand are essentially protons and have a partial positive charge associated with them.

Sugar is also polar! It has many hydroxyl (OH) groups that readily make hydrogen bonds. Sugar thus has both partial positive and negative charges associated with it. As a result, there are hydrogen bond donors and acceptors in both water and in sugar. For this reason, sugar will dissolve in water.

On the other hand, something like oil is made up of primarily of C-H bonds. As discussed above, a C-H bond is not polar because the electronegativity between the two atoms in the bond is not that different. This means that overall, oil doesn't really have any sort of partial positive or negative charge. This lack of partial charges means that the oil molecule will not be able to hydrogen bond. Since water likes to hydrogen bond and stay with polar molecules, the water will not dissolve the oil.

Taking a look at the structure of the compound and the nature of the bonds it contains will tell you a lot about whether or not the molecule can have partial positive or partial negative charge. If it can, then it is likely polar. It if does not, then it is nonpolar.