Trees swayin' in the summer breeze
Showin' off their silver leaves
...
Um, wait a moment, leaves are green why are the leaves in these pictures silver?!

Background

Light that bounces off of non-metallic objects (this includes the dust and water in the air) becomes polarized. With light that bounces off a subject, the maximum polarization occurs when at Brewster's angle or about 56 degrees from the normal (34 degrees from the surface). This depends upon the subject, water is at 37 degrees, and glass is at 32.

When the polarizer is rotated to a non-polarizing angle (or is taking a picture of a completely unpolarized source such as a lightbulb), it has a filter factor of about 2.5 (1.25 f/stops of light are blocked)

Silver Leaves, Water, and Glass

Light bouncing off the leaves or water produces a glare that may not be the image desired. A photograph of lily pads on water may turn out with the leaves and water both being a shiny gray rather than green leaves and clear water. Adding the polarizer will block the glare from the leaves and water producing green leaves. Blocking the glare can help with photographing all trees and greenery producing a more saturated green than would otherwise be captured on film.

The top surface of the water reflects back polarized light. When taking pictures and wanting to capture the image of what is under the water (stones, sand, fish (fishermen often wear polarized sunglasses for this reason - how else can they tell of the big one that got away if they didn't see it?)). Do realize that it may not always be the best thing to remove these reflections. Often the highlights of the waves can add some extra power to the photograph that wouldn't be there if a polarizer was used.

The polarization also is applicable to glass. For this reason, drivers often wear polarized sunglasses to reduce the glare from the windshields of other cars (and their own). When taking a photograph through glass, the polarizer filter will remove the reflections in the glass allowing the subject behind the glass to be seen clearly.

Blue Sky

With respect to the sky, there are two factors at work: Rayleigh scattering (scattering of light from gas molecules) and Mie scattering (scattering of light from dust and water). Each of these types of scattering polarizes the light.

Our eyes are amazing things allowing for handling several orders of magnitudes different levels of light to be resolved. This biases us to seeing the sky as blue when we look at it. Look at a landscape and the sky... and that blue sky is actually about 2-8 times brighter than the landscape. We have no problems with this.

Film on the other hand does have a problem with too many stops of contrast between the brightest bright and the darkest dark. When taking a picture and metering the land, we get a sky that is light blue or near white (overexposed). If the sky was metered, the land would be too dark (underexposed). This is most prevalent in slide film where the film can not handle too much dynamic levels. If only there was some way to darken the sky without darkening the land...

In black and white, this would be done by using a filter that is about 120-180 degrees off of the color on the color wheel. To make the sky darker with black and white photography, one uses a red or red-orange filter. This doesn't work with color film (unless you want the entire picture to be red) and thus the polarizer is used.

The greatest polarization of the sky will occur 90 from the sun - with the sun out of the corner of your eye rather than behind or in front of the camera. The "rule of thumb" applies here: make an 'L' with the thumb and index finger. Point at the sun, and the greatest polarization will be the thumb points (the hand may be rotated, as long as the finger remains pointing at the sun). At noon, the horizon in all directions is the most polarized.

Artwork

Photographing artwork can benefit from the polarizer. As with leaves and water above, light bouncing off the artwork to be photographed may be seen as glare. This may be corrected by moving the light and/or the camera, but this is not always an option (sometimes the light and camera are best where they are). To eliminate the glare, two polarizing filters are used - one on the light source and the other on the camera. The two polarizers are to be 90 degrees offset from each other. This is known as cross polarization.

Circular Polarizer?

With modern SLR cameras, there is a beam-splitting prism that diverts some of the light to the light meter. This beam splitter will partially polarize the light. If the light entering is already polarized this will reduce the light reaching the light meter even more than it would be otherwise resulting in an overexposed image. To get around this problem, there is a thin layer added that is a 1/4 wave retarder essentially depolarizing the light after if has been filtered. This foil twists the light so that it appears unpolarized to the meter and no light is lost there. This filter isn't needed on a digital camera because the light reading and the image capture is often the same thing, though a circular polarizer may also be used without problem.

To determine if a polarizer filter is circular or linear, look at it in the mirror. If the filter is clear, then the filter is linear. If the filter is dark, the filter is circular.

Other considerations

When using a wide angle lens with a polarizer two factors come into play. The first is that the sky is not evenly polarized and it will become apparent that there are be a band of dark sky with lighter sky on either side. This is most easily captured by wide angle lenses. The other thing to consider is that the polarizer is often a thick filter (needing the mounting to be able to rotate). If one was to stack the filter, some mechanical vignetting may become apparent as a dark fuzzy ring on the corners. For this reason, polarizers are available with the two most useful (outdoors) variants that may be stacked: warm tone (skylight) and red enhancing. Some manufactuers also have thin mounts (the B+W slim mount for the circular polarizer has a height of 5mm though is only available in ranges from 49-77mm (normaly goes up to 127mm) and costs an arm and a leg more).

Digital Darkroom

The polarized image is one that is best done in camera - before the image is recorded. While it is possible to sit down with Photoshop, select the entire sky and then play with the levels, this is much more difficult to do trying to change the silver leaves to green. As a film photographer, I have a bias to doing everything I want to prior to the film being exposed that is lost on many digital photographers. There are some things that are just not doable with Photoshop (or even in the chemical darkroom) after the photograph has been recorded - removing reflections from glass, properly saturating the greens and darkening the blues are just some of them. These are things best done with the proper tool - the polarizing filter.

http://webexhibits.org/causesofcolor/14B.html
Ansel Adams, The Negative
B+W Promotional Material
http://www.tiffen.com/polarizer_pics.htm
http://scienceworld.wolfram.com/physics/BrewstersAngle.html