Phase contrast microscopy is a technique used when a high magnification is desired of a primarily translucent specimen. When looking at many types of cells under a standard brightfield microscope it is extremely difficult to make out many fine details of cellular structure because many of organelles are virtually clear. While the cells do not block any of the light, typically the various cell structures will at least shift the phase of the light waves passing through them. Phase Contrast Microscopy allows for better viewing of these specimen by converting differences in refractive indicies and thickness into a visible sprectrum of light and dark.

How it works:

Traditional Brightfield microcopy involves high intensity light either shining directly through the specimen (transmitted brightfield), or shining from above and reflecting back up (reflected brightfield). In either case, contrast arises from the differences in opacity across the specimen:

Example of Transmitted brightfield:
           | Lens |
           |      |     3: Light enters lens and specimen
            ======         is seen
             \  /
             /  \  
            /    \ 
          (specimen)    2: Some of the light is absorbed
        =============      or reflected yielding contrast
         /          \
        /            \      1: Focused light leaves 
       (==============)        Source
        |Light Source|

Phase contrast instead blocks out all light except for a ring using a condenser annulus, which is effectively an opaque plate that has the outline of a circle cut out so that only a ring of light gets through. The ring of light is then focused onto the specimen, various parts of which will bend or slow down the light to varying degrees. This diffracted light then passes through a phase plate, which is typically a plate of glass with a circle etched in it. The phase plate is where the diffractions are amplified in such a way that there is contrast for the image:

Example of Phase Contrast:
           |  Lens |
           |       |    5: Light enters lens and specimen 
            =======        is seen
            |.....|     4: Light passes through phase plate
            =======        and refractions are converted
            \..|../        into contrast and refocused.
          (specimen)    3: Some of the light is bent or 
         =============      diffracted up into the objective
          //********\\   2: Condenser blocks all but a ring
         (============)     of light which is focused on 
          |          |      the specimen
          |          |
         |Light Source|  1: Light leaves light source
Why it works

The tendency of light to spread out after passing through a slit is known as diffraction. In this case the condenser acts as a circular slit. As the light passes through the specimen much of it continues unhindered and provides the background, sometimes called the surround wave. Some of it will be diffracted when it passes through a medium (i.e. your cellular structure) with a higher refractive index than its surroundings (which is generally air or immersion oil). This set of light is often called the diffraction wave. These two waves interact with each other similar to the waves in the double slit experiment, and depending on how much the diffraction wave was shifted relative to the surround wave, interference will occur that particular spot will appear darker. In order to remove the multiple modes of the diffraction pattern from the final image, the light is passed through the phase plate which slightly shifts the surround wave forward for only one of the modes. This shift amplifies the interferece between the diffraction and surround waves, increasing the final contrast in that mode. The net result is that there is a sharp image where parts of the specimen that are thicker or have a higher refractive index (typcially organelles) appear darker*, and spots with a lower refractive index (such as membranes and cytoplasm) appear lighter. Additionally many fine structures that would not be well defined under brightfield will appear much sharper under phase contrast.

Phase Contrast Microscopy is based on Frits Zernike's work with diffraction gratings. In 1953, Zernike was awarded the Nobel Prize for Physics for his work with phase contrast.

* This is assuming positive phase contrast is being used. There is also negative phase contrast where the surround wave is shifted backwards instead of forwards. This ends up inverting the phase contrast effect so that lower refractive indices will be darker instead of higher ones.

For more info check out:
MicroscopyU by Nikon:

Zernike's Nobel Lecture: