Polarization is a phenomenon peculiar to transverse waves.  Longitudinal waves such as sound cannot be polarized.  Light and other electromagnetic waves are transverse waves made up of mutually perpendicular, fluctuating electric and magnetic fields.  In the diagram below an EM wave is propagating in the x-direction, the electric field oscillates in the xy-plane, and the magnetic field oscillates in the xz-plane.  A line traces out the electric field vector as the wave propagates.

An unpolarized electromagnetic wave traveling in the x-direction is a superposition of many waves.  For each of these waves the electric field vector is perpendicular to the x-axis, but the angle it makes with the y-axis is different for different waves.  For unpolarized light traveling in the x-direction E_y and E_z are randomly varying on a timescale that is much shorter than that needed for observation.

Unpolarized light: Natural light is, in general, unpolarized.

For a linearly polarized electromagnetic wave traveling in the x-direction, the angle the electric field makes with the y-axis is unique.

An ideal polarizer is a material that passes only EM waves for which the electric field vector is parallel to its transmission axis.  The electric field is a vector and can be written in terms of the components parallel and perpendicular to the polarizer's transmission axis. 
E = E_parallel + E_{perpendicular}. 
An ideal polarizer passes E_parallel and absorbs E_{perpendicular}.

If E₀ is the incident field vector and the angle between E₀ and the transmission axis is θ, then the magnitude of transmitted field vector is E₀ cosθ and its direction is the direction of the transmission axis.  The intensity I of an electromagnetic wave is proportional to the square of the magnitude of the electric field vector.  We therefore have

I_transmitted = I₀ cos²θ.

This is called the law of Malus.  If θ = 90^o the transmitted intensity is zero.

The lines indicate the direction of the transmission axis.

When unpolarized light passes through a polarizer, the intensity is reduced by a factor of ½.  The average of cos²θ, averages over all angles θ is ½.
I_transmitted = I₀<cos²θ>_{all angles} = ½I₀.

Problem:

A beam of unpolarized light of intensity I₀ passes through a series of ideal polarizing filters with their transmission axis turned to various angles, as shown in the figure.
(a)  What is the light intensity (in terms of I₀) in regions A, B, and C?
(b)  If we remove the middle filter, what will be the light intensity at point C?

Solution:

• Reasoning:
  When unpolarized light passes through a polarizer, the intensity is reduced by a factor of ½.  The transmitted light is polarized along the axis of the polarizer.
  When polarized light of intensity I₀ is incident on a polarizer, the transmitted intensity is given by I = I₀cos²θ, where θ is the angle between the polarization direction of the incident light and the axis of the filter. The transmitted light is polarized along the axis of the polarizer.
• Details of the calculation:
  In this problem we have 3 polarizing filters.  For the second polarizer θ = 30^o between the polarization direction of the light incident of the filter and the axis of the filter.  For the third polarizer θ = 90^o - 30^o = 60^o between the polarization direction of the light incident of the filter and the axis of the filter.
  We then have that:
  (a) In region A the intensity is I₀/2 and the light is polarized along the vertical direction.
  In region B the intensity is (I₀/2)cos²30^o, = 0.375 I₀, and the light is polarized along the axis of the second polarizer.
  in region C the intensity is (0.375 I₀)cos²60^o = 0.0938 I₀ and the light is horizontally polarized.
  (b) If we remove the middle filter, for the last filter we now have that θ = 90^o.  Thus I = 0.
  It is important to visualize the fact that adding the middle filter increases the transmitted intensity!
  This "paradoxical" effect is a signature of wave phenomena in general.

There are different polarization mechanisms.  The most common method of producing polarized light is to use polaroid material, made from chains of organic molecules, which are anisotropic in shape.  Light transmitted is linearly polarized perpendicular to the direction of the chains.  The transmission axis is perpendicular to the chains.

A polarizer produces linearly polarized light.  It is often convenient to orient the transmission axis of a polarizer vertically or horizontally to produce light with vertical or horizontal linear polarization.

Vertical and horizontal polarization

Polarization by reflection:

When unpolarized light is incident on a boundary between two dielectric surfaces, for example on an air-water boundary, then the reflected and transmitted components are partially polarized.  The reflected wave is 100% linearly polarized when the incident angle is equal to an angle called the Brewster angle. 
For water this angle is is ~53^o with respect to the normal or 37^o with respect to the water surface.
For are considerable angular range around the Brewster angle the reflected light is highly polarized in the horizontal direction.

When the sun is at a low angle in the sky, the sunlight reflecting off the surface of water is nearly 100% horizontally polarized because the angle of incidence is close to the Brewster angle.
Glare-reducing sunglasses are coated with a polarizer with a vertical transmission axis and therefore block the reflected light.

External link:  Polarization of light (Youtube)
External link:  Polarization and reflection (Youtube)

Demonstrations

The way light is transmitted through some materials depends on its polarization.

Certain birefringent crystalline substances bend light trough an angle that depends upon the state of incident polarization.  The have an optic axis.  Unpolarized light entering a birefringent crystal not along the optic axis of the crystal is split into beams which are bend by different amounts.

Double refraction in a calcite crystal.  The vertical lines are imaged twice.
The polarization of the light producing one image is perpendicular to the
polarization of the light producing the other image.

Some materials turn birefringent when stressed.  By placing transparent materials between two polarizers, we can perform stress analysis tests.

Optically active or circular birefringent materials rotate the direction of polarization of linearly polarized light.  The amount of rotation depends on the wavelength of the light.  Sugar molecules have a handedness (chirality) and in solution are optically active.  If we polarize white light and pass it through sugar syrup, the direction of polarization of the light emerging from the syrup will be different for the different color components.  If the light then passes through a second polarizer, its color changes with the orientation of the transmission axis of this polarizer.

Sugar syrup between crossed polarizers
with different relative orientations