Making spore prints can be extremely helpful when identifying wild mushrooms. While one can't see a single spore, the color of a large group of spores often gives a clue towards the type of mushroom.
Spores are the "seeds" of the mushroom, dropped or released with the intention of reproduction. For more on spore dispersal methods see this page.
The Dartmouth Electron Microscope Facility at Dartmouth College kindly released the adjacent picture into the public domain. This is a close up of Agaricus bisporus spores using an electron microscope. We'll need to see a print of these en masse in order to determine their color with the naked eye!
Spore prints alone aren't enough to make a positive identification but their color can be an important trait to note.
After the print is made compare the color to the description in your guide books. Some color variations may seem objective (how many shades of white are there, anyway?). However, brown from pink should be easy enough to determine.
Below is a list of just some mushrooms and their known spore colors:
Certain wild mushrooms, morels for example, don't have obvious spore producing tissue such as gills or pores. Simply use the entire mushroom here and expect to see spores spread loosely around your specimen, not in a distinct pattern.
The second picture was taken by Eric Guinther and is published on Wikipedia under the GNU Free Documentation License.
The third picture was taken by Ranko and is published on Wikipedia under the Creative Commons Share Alike 3.0 License.