To expand on what geepondy said...
The aperture (F-stop) controls how much light gets to the film/sensor in a camera. The higher the number, the less light gets in, and the longer the exposure.
Changing your aperture by a "full stop" means changing it so your exposure doubles or is cut in half.
For instance:
If you shoot a picture, and your camera says: 1/60, F2.0
Changing the aperature to these value will cause the exposure to be affected as such:
<font class="small">Code:</font><hr /><pre>
F-stop Exposure (seconds)
2.8 1/30
4.0 1/15
5.6 1/8
8.0 1/4
</pre><hr />
So F/8.0 lets in 16 times less light, so using that setting, the exposure must be 15 times longer. If you change your camera from F/4.0 to F/5.6, your exposure will be twice as long.
If you need a specific shutter speed, you can adjust the aperture to get the proper exposure.
The other thing that aperture affects is depth of field - which is how much is "in focus". The higher the F-stop number, the larger the depth of field, so more will be in focus.
So, let's say you take a picture of something, using F/2.0, and the depth of field ends up being 1 foot. That means that everything from 6" in front to 6" behind whatever you focused on will be considered "in focus" (it's not really split 50/50 like that, but we'll skip that for simplicity).
So increase the F number to 8.0. Now your depth of field may be 5 feet (depends on the lens and lots of other factors). So everything from 2.5 feet in front of, to 2.5 feet behind what you focused on will be considered "in focus".
Finally, there is usually a "sweet spot" for sharpness, that's generally around halfway between the ranges of F-stops the lens will do. On my camera, the range is F/2.0 - F/8.0. The sweet spot is at F/5.6. Things are a touch more blurry and you get some color artifacts at F/2.0, and a touch blurry at F/8.0. At F/5.6, my camera lens is providing optimal image sharpness.
If you're trying to achieve some specific effects with your picture, you can use aperture to do that. If you want a lot of the picture in focus, use a high F-number (like F/8.0). If you want just your subject in focus, but the background blurred, use a low F-number (like F/2.0).