Normal light waves go in every direction. In a polarizing microscope the light waves are managed using two polarizing filters. One is called the analyzer and the other is the polarizer.
One way to understand how to create polarized light is to pass ordinary light, containing light waves that are vibrating in all directions, through thin slots cut in a piece of paper.
If the slots were aligned vertically then only the vertical light waves would pass through while the others would not. The vertical lights coming out of the slots are considered polarized light waves. These light waves in turn then illuminate the sample, usually crystals of some type. After passing through the crystal they are now out-of-sync because they moved at different velocities within the mineral. Once the light waves pass through the analyzer the colorations of the crystal provide identification of such samples.
Good examples of polarizing microscopes would be the Meiji ML9420 or the VanGuard 1242MM.
Most minerals are viewed where these polarizing filters are turned at 90 degree angles to each other. Polarizing microscopes are used to observe minerals and crystal formations. Applications can be in the clinical field for gout, law enforcement for drug analysis, industrial testing and geological applications.