This one is pretty basic. If you are going to be out shooting in nature, this is your single greatest tool.
What a CPL does is remove unattractive reflected or glare light. When light becomes reflected from a surface, it becomes polarized. By rotating the filter, you can block this light. This is how photographers remove reflections off of water, give foliage a more saturated feel, and create deep graduated skies.
In the example of the water running over the rocks you can distinctly see the effects of a CPL. Keep in mind that a CPL when set to block reflections reduced on average one stop of light. Both of the images were shot at the same settings at show this loss of light in addition to removal of reflection.
The CPL darkens the sky creating a strong and apparent graduation of tone (which is not only aesthetically pleasing, but can help create the illusion of a greater dynamic range) but this can be drawback in some cases, especially when shooting with a wide lens. In the example image you can see that the CPL creates a giant unpleasant wedge shape of tone in the sky. The wedge becomes apparent at wider angles. Keep in mind though, the angle of the graduation of tone is related to where the sun is in relation you. In the image the sun is rising and I am shooting the northeast. If I had been facing west, the sky graduation would go up and down and not an angle.
There are other drawbacks to using CPL, but they can be ameliorated somewhat by simply buying a nicer CPL. The problems will always be there, but the higher quality filters will suffer from it less and in less conditions. Cheaper filters can cause your images to be slightly blurry.
The worst problems appear when the sun shines directly onto your filter. Two things can happen; contrast loss and flare. When the sun shines onto your lens some of it is reflected off the filter, washing out the image. When the sun hits imperfections and the like or becomes split, it can flare. See the example image for flare. More expensive filters try to prevent this with different coatings and lens materials. Keep in mind that is may not be just the filter causing this, but the lens itself. Lenses like filters suffer from these problems.
Flare and contrast can be fixed in photoshop, but its better to get it right before hand. Every edit you make degrades the image slightly. Another two edits can make or break a sky from being a nice gradient to a banded mess of garbage.
A rare but terrible problem that can happen is when light reflected off the lens itself on the filter becomes strong enough to be visible in the image. This occurs under very certain conditions. I have only had it happen to me with CPL a handful of times out of the 20,000 or so shots I have made up to this point.
The bottom line is, buy a CPL as a tool it is essential for photographing nature even with its drawbacks and problems.
All the images presented with this are unedited raw images resized and labeled
Next will be Cyan, Cobalt blue, and You messing with the Hue