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A neutral density filter or ND filter is a filter that reduces the intensity of all wavelengths of light equally. ND filters do not change the hue of a color rendition.
Why would you use an ND filter? Well generally speaking you use them when you want to lower the intensity of light in a situation. Why would you want to do this? Simply put, if a scene is too bright you can slap on an ND filter and you now have a much less bright scene.
Have you ever wanted to show motion or get a nice blue effect? ND filters are great for this. You know those photos with blurred out waterfalls like this:
photo by vinothchandar
These are generally shot with an ND filter. By using the ND filter you can shoot much longer exposures because there is less light entering through your lens.
There are various types of ND filters. There are the screw on type where you match the size and threading on the front of your lens to the filter. There are also square bracket mounts and that you screw onto your lens. Basically you mount a holder onto your lens and then just slide the filter in. This system is nice because you don't have to have the exact same filter size as each lens you have.
Here are some more great videos on this topic:
You can check prices and buy ND filters here.
ND Filters Wikipedia.
More photography resources.
Photography glossary.