Polarity is very difficult to teach to my sophomore chemistry students. I have two "go-to" demos that I like to use to introduce the concept of polar and nonpolar molecules.
Polar molecules, like water, will be attracted to an electric charge while nonpolar molecules are unaffected by the charge. To bring this difference in physical property to life, I prepare two burets for the demo, one containing water (with a little blue food coloring to make it easy to see) and the other containing heptane (any nonpolar solvent will work). I borrow a plastic stick and piece of rabbit fur from the physics closet to generate a negatively charged object. In class, I open the stopcock for both burets, charge up the plastic stick with the rabbit fur, and pass the charged stick near the two streams. When I pass the negatively charged stick by the stream of water, it bends toward the stick. The challenge is to get the water to bend enough to hit another beaker. The stream of heptane is unresponsive to the charged object.
An interesting observation happened the last time I tried this demo for my students. The charged stick, as expected, easily bent the stream of water. The stream of heptane was not responsive to the negatively charged stick, at first. But when I tried it another time (the cry of “do it again” is impossible to resist), the stream of heptane bent toward the charged stick. This observation was completely unexpected. I actually exclaimed, "That's not supposed to happen.”
I tried it again, and again the heptane stream was attracted to the stick. What we found is that if I passed the stick near the tip of the buret, the heptane stream would bend. But, if I approached the stream farther down, away from the opening of the buret, there was no response to the stick. I tried a second batch of fresh heptane, same result. So I took the heptane and tried to dissolve some food coloring into it. In this case, the heptane behaved as any nonpolar substance; the food coloring did not dissolve. My best explanation of this anomaly is that, because the conditions were very dry on the day of the demo, the heptane gained an induced charge from the stick, which then allowed it to be repelled by the charge.
Chemists often use the saying "Like dissolves like." In the second “go-to” polar/nonpolar demo I show my students how this phrase plays out. I pour about 40 mL of water and 5 mL of heptane into a eudiometer tube (any nonpolar solvent will work here.). Then I put a small pellet of iodine, a nonpolar solid, in the tube. As I slowly invert the tube several times, the heptane layer takes on a beautiful pink color from the dissolved iodine, but the water layer stays clear. The nonpolar iodine dissolves only .in the nonpolar solvent. 
To make it even more interesting, I drop some food coloring into the tube. Food coloring is a water-based solution, so it will not dissolve in the nonpolar heptane layer. Telling the students this is one thing, but seeing it, in this case is really believing. The food coloring stays in a tight drop as it travels through the nonpolar layer and leaves no trace of interaction in the heptane. But once it hits the water layer it begins to spread out into the water layer. It is very cool to watch.
Using these two demonstrations help to attach a concrete physical observation to the abstract concept of molecular polarity. Understanding the impact of polarity on the properties of a substance is one thing, but getting my students to identify which molecules are polar and which molecules are nonpolar based on the structure is not as easy.
[Sharon has a great blog called The Art of Teaching Science.
Note: Dispensing heptane should be done in a fume hood since it is a volatile liquid. Please review the MSDS information as heptane is flammable and may also be considered toxic. Teachers should remember that iodine is hazardous and limit the amount used — only a very small pellet is needed for the desired effect. The inorganic and organic waste should be separated for proper disposal.]
