What is a variable neutral density filter?
The neutral density bit means it is a filter simply designed to block some of the light getting into a camera. The variable bit means it is variable - you can control the darkness of the filter just by twisting one part of it. A proper variable neutral density filter can cost £100 or more!
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!