Let’s start with the generic definition provided by Wikipedia.

In photography and optics, a neutral density filter or ND filter can be a colorless (clear) or grey filter. An ideal neutral density filter reduces and/or modifies intensity of all wavelengths or colors of light equally, giving no changes in hue of color rendition.

The purpose of standard photographic neutral density filters is to allow the photographer greater flexibility to change the aperture, exposure time and/or motion blur of subject in different situations and atmospheric conditions.

Basically a Neutral Density Filter (ND Filter) can be used in almost an unlimited amount of ways in artistic photography. First and foremost is the coveted daytime long exposure. By adding a ND Filter to your lens, it will allow you to expose for a longer amount of time, thus adding motion blur to any object that you are shooting. Cue the classic waterfall shots with soft, velvety water that just screams computer desktop background image!