The main feature of ND filters is their ability to give you control on the shutter speed.
ND filters come in wide range of values, from ND2 to ND100000, but can be generally summarized in 3 main categories: light, middle and heavy.
"Light" ND filters are generally used in portrait photography; "middle" ND filters are commonly used in landscape photography and "heavy" ND filters prove their best in "imaginary landscape" or "Another World" style of photography.
"Light" filters can be used in various ways in portrait photography, and in day light conditions can be used with F1.2 or F1.4 fast lenses. One effect usually aimed in this photographic genre is a beautifully blurred background, easily obtained with F1.2 or F1.4 fast prime lenses at maximum aperture, but to do so a very high shutter speed is required. However, depending on the model, some cameras come with a maximum shutter speed of 1/4000 sec that, when shooting with F1.2 or F1.4 prime lenses, may cause over-exposure. Nevertheless, it is possible to control the shutter speed by using "light" ND filters, for example in ND2 or ND4 values, to shoot with appropriate exposure.
"Middle" ND filters can perform motion control and are therefore used to render the movement of moving objects such as water flow, wave motion, people flow, etc. As a well-known example, when shooting at a fountain without ND filters the fountain's water appears "still" on the final image. In daylight conditions, one way to avoid this effect and have the fountain "flowing" on your final image (but still without using ND filters) will be to extremely reduce the aperture to a value of f/16 or f/22.
However, when shooting at such reduced aperture value, "diffraction" may occur, making resolution poor, and the overall depiction as poor as well. Also, when cameras with ISO200 lowest sensitivity and fast prime lenses are used together, the exposure may not be reduced enough to that shutter speed at which the "water flowing" effect can be expressed. However, by using "middle" ND filters, it is possible to shoot at a proper F value to avoid "diffraction", and it is possible to lower the exposure to that shutter speed value necessary to express the water "flowing".
"Heavy" ND filters are particularly used when shooting "imaginary landscape" or "Another World" style of photography. With these filters, you can make a person "disappear" from a cityscape, shoot in long exposure under daytime light conditions, or take time lapse shooting of clouds and create a movie of fascinating and otherworldly fast clouds streams. These filters can be also used to capture a total solar eclipse with great results.
As a trick with ND filters, light may enter from the viewfinder when performing long-time exposure in daytime light conditions, so it will be necessary to close the viewfinder when using these filters. With cameras equipped with viewfinder shutter, the viewfinder must be closed to shoot; with cameras without viewfinder shutter, you will need to set the eyepiece cap before shooting.