DOF stands for "Depth Of Field" so a DOF preview shows you what sort of depth of field you're going to get in your photo.
When your camera locks focus on a subject, that spot is called the plane of focus.
There is an area in front of and behind the plane of focus that ALSO will appear in focus and this is the depth of field.
Depth of field is controlled by the aperture of the lens - a wide lens aperture results in shallow depth of field while a narrow lens aperture results in deep depth of field.
Shallow depth of field is commonly used to blur the background in a portrait image while deep depth of field is used to keep everything sharp in a landscape photo.
A depth of field preview option on a digital SLR camera lets you "see" the depth of field in your photo by looking through the viewfinder.
When you press the DOF preview button, the camera sets the lens aperture to whatever value you have selected (usually the lens is wide open).
If you select an aperture of f/16, then pressing the DOF preview button will narrow the lens aperture to f/16 and you should be able to see lots of depth of field when you look through the viewfinder.
The only drawback is that when you narrow the lens aperture, you also dramatically reduce the amount of light passing through the lens and reflecting up to the viewfinder.
This is why you will often see the image in the viewfinder get MUCH darker when you use a DOF preview.