Why would you want to block light getting into the camera? In short; control. A fully manual camera can be controlled via exposure time, aperture size and film speed. Adding a variable neutral density filter adds control of the amount of light entering the lens too. This lets you increase the exposure time and/or aperture while using the neutral density filter to prevent overexposure.
What effects are possible? The main tricks with a neutral density filter are to get shallow depth of field (a wide aperture) or long motion blur (a long exposure) under bright lighting conditions. This makes it very handy for taking portraits or nature shots, where you have bright lighting but want a shallow depth of field, or capturing the feel of a public event while bluring out individual people as they move.
How does it work? This method uses the properties of polarised light, specifically that two parallel polarisers will block very little light but two at 90 to each other will block nearly all light travelling through them. Find out more about polarisation, including in photography, here. This variable neutral density filter is far from perfect, but great if you want to make one cheaply!
Step 1: Raw Materials
Step 2: Dissasemble One Filter
Be careful with this step - it is very easy to slip and scratch the filter. Make sure you don't do this with a polariser you care about!
Step 3: Flip the Filter and Reassemble
It is easy to check the correct orientation of a circular polariser if you get confused. Find a polarising light source, most LCD screens should work, and look through the filter while twisting it. For the normal orientation of a circular polariser if the filter gets darker and lighter then you are looking through the camera-facing side of the filter. If filter stays the same lightness but changes colour slightly (normally yellow to blue) then you are looking from the other side of the filter. For the reversed orientation of the circular polariser (for this step) the opposite applies.
Make sure you mark which filter has been flipped so you don't get confused!
Step 4: Final Assembly
If you don't plan on using the unmodified filter as a polarising filter on it's own then you can glue the two together to make sure you don't get confused. If you do want to glue the filters together be careful; to use the filter you have to be able to twist the front filter while the back filter remains stationary - don't jam the twisting mechanism with glue! Also avoid using cyanoacrylate-based glues (eg. superglue and krazy glue), their vapours can fog the glass.
Step 5: Usage
The light from the filters entering the camera is circularly polarised so should work with all digital camera autofocus and metering mechanisms. Unfortunately, because this method is based on polarisers, you will see some of the normal effects of polarising lenses - bear this in mind if you are photographing reflective objects such as glass or water. The filter construction is also quite thick so you might get more vingetting, especially at short focal lengths on zoom lenses.
Depending on the quality of the filters you might see some colour changes depending on the orientation of the filters, blue in one direction and yellow in the other. The blue tint can normally be countered with a fluorescent light white balance setting.
Step 6: Applications
A flower under bright sunlight - use a wide aperture and a dark neutral density filter to capture the flower, without overexposure, with a nicely blurred background.
People in movement - use an extremely long exposure time and a very dark neutral density filter to blur the movement of people through a public space.