MAIN APPLICATIONS - ND FILTERS ALLOW:

1. A larger aperture and consequently a smaller depth of field which helps separate subjects from their background (which becomes blurred.) The lens operates then in the range of its best optical performance, say around f/5.6 to f/8 rather than a very small opening where diffraction or other optical artifacts might become an issue.
2. A much slower shutter speed to be used at reasonable aperture, to record movement in subjects such as waterfalls, clouds, cars, seascapes etc.
3. A decrease of the effective ISO of high speed film (above ISO 400) to make it possible to be used outdoors in bright situations.
4. A proper use of cine and video cameras (which have fixed shutter speeds) to film subjects such as snow, sand or other bright scenes which often cause over exposure issues.
5. A closer focus and larger depth of filed in flash macro photography without overpowering the image.

ND filters come in many strengths or different levels of densities
for fine tuning exposures as well as various rates of transition:

ND2 reduces amount of light to 50%
ND4 reduces amount of light to 25%
ND8 reduces amount of light to 12.5%

in other words, ND-2 absorbs 1 F-stop of light.
ND-4 absorbs 2 F-stops of light.
ND-8 absorbs 3 F-stops of light.

Examples on right - top without filter. Bottom with Cokin ND-8.

Why graduated Neutral Density?
Unlike human eyes, CCD or film have no emotions or imagination. THey only see light as levels of brightness. A large difference in brightness between sky and foreground is beyond their ability to see and record. This is where graduated neutral-density (ND) filters come in. By using one of these filters, you will be able to bring to CCD or film what you can see with your eyes. Grad ND filters can't create light but they can help you capture it properly.
Pro landscape photographers often use Grad ND's to darken a bright sky so that both the sky and subject can be properly exposed, thus capturing greater detail.
Use these filters when you need to control the contrast range in your landscape images to avoid blowing out highlights like snow-covered mountains or dramatic skies.
How do you know when you need one? With some experience, you get a pretty good idea of when the light range in a scene is beyond the ability of the CCD or film to record, but one sure way is to use your in-camera spot meter and evaluate the scene. Check the difference between the exposure needed for your foreground and the exposure needed for the background. If there's no difference between the two, then you do not need a filter. If there is, say, a 2-stop difference, then you need a 2-stop filter (ND-4).

Several Types:
Cokin Graduated ND filters are available in "soft step" and "hard step" types. The soft-step filters are suited for scenes with no distinct boundary between light and dark zones. The hard step are used for scenes where sky and foreground areas are distinctly separated, usually at the horizon. Soft-step filters are however the universal choice. Both the soft-step and hard-step types are offered in the 3 major density graduations (1 / 2/ 3 steps).
To be noted: wide-angle lenses, because of their inherent greater depth of field (particularly with smaller apertures), will deliver a more pronounced and noticeable graduation than higher focal lengths (50mm and up).