The F/stop on a camera controls the lens aperture: the smaller the number, the bigger the aperture and the more light it allows in.  You calculate it by divding the size of the lens by the diameter of the aperture: so for a 50mm lens, an aperture 25mm wide will be an f/2, a 12.5mm aperture will be f/4, and so on.

If you reverse it so you divide the lens size by the f/stop, you get the size of the aperture.  Knowing the diameter, you know the radius and thus can calculate the area of the aperture using the old radius multiplied by pi R-squared equation that everybody should have learned in school.  Again using a 50mm lens, f/2 gives you an aperture of 25mm (radius of 12.5mm), and an area of about 491 square millimeters.  Setting it at f/2.8 gives a aperture of 17.9 mm and an area of about 250 square millimeters, or roughly half the size.

The amount of light gathered by the film/CCD/virtual thingie that calculates the rendered image is directly proportional to the area of the aperture: twice as much area, twice as much light gets in, which means that everything you see gets twice as bright as it was.  If the settings were already set up to give you a good range of illumination in the rendered image from the brightest to the darkest portions, twice as much light will blow out the parts that were toward the upper end to begin with.

The joy of the f/stop is that it doesn't matter what size lens you have on the camera: since it's the ratio of lens size to aperture, you know that setting the lens to f/2 will always allow in about twice as much light as f/2.8  It's also the reason why f/stops have been standardized on the vast majority of cameras to those odd numbers you see: each setting is roughly twice the area of the next highest number, and thus, everything else being equal, the exposed image will be twice as bright overall.  So f/1.4 is twice as bright as f/2, which is twice as bright as f/2.8.