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!
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In looking around online, I've seen both methods of making a variable ND - two circulars or a circular and a linear. However, using the linear/circular method seems as if it does not suffer from that "blue hue" curse (check this video for proof: http://www.youtube.com/watch?v=Jmwpi0RRuyo). I purchased a cheap 77mm linear and circular off eBay and when trying to combine the two, I only get a very weak darkening. Nowehere near the total blackness you see in that video, or with two circulars.
I managed to get my hands on another linear polarizer (unfortunately, it's a 49mm, so it's useless to me) and when I pair that one with my circ, it works! Looking at my 77mm linear package, it actually calls it an "SR Polarizer." I've found only a few uses of the term "SR polarizer" online and none that seem to dictate any difference between SR and linear.
Anyone got anything on this?