a neutral density filter whose density (that's to say, degree in which it blocks light) is not constant, but varies across its surface.
If you have never seen one, think of those mildly cheesy sunglasses that are darker on the top part of the lens, and nearly clear on the bottom. Got it ?
What is the application of such a filter ? It is used, in photography, for taking pictures of scenes where the contrast range exceeds what the film can record.

For example, consider the following scene:

  cloudy sky
   *     o
  \|
   |/ ...
------------
##hearth####
############

The funny looking thing is a tree.
Let us meter the scene, for an aperture of f 16:

1/1000
   *     o
  \|
   |/ ...
------------
##1/60#######
############

The problem is that between 1/1000 and 1/60 of a second there are 5 stops, which exhausts the contrast range of slide film. This means that if we expose at, for example 1/250, the ground will be completely dark and the sky completely blown out, i.e. white.

In order to rescue our career of landscape photographers, we need a way of dimishing the amount of light from the sky. Enter the graduated neutral density filter, normally in the form of a square piece of glass or resin in a sliding Cokin mount:

++++++++++++++
++this part+++
++absorbs+++++
++3 stops+++++
--------------
|            |
|this part is|
|transparent |
+------------+

If you superimpose this filter to the above scene, you will notice that the sky goes down to 1/125. If you expose at 1/60, the sky will be at +1 stop, which allows for plenty of detail in the clouds, and the ground will be perfectly readable.
Graduated ND filters differ on the basis of how sharp the transition between the light and the dark part is; a sharp transition works well with the horizon and buildings, while a soft transition agrees with fuzzy things, like trees.
Avoid 1 stop graduated ND filters, because they are nearly useless. I find myself using mostly the 3 stop filter, and wishing for a 4 stop one at times.

There are also color graduated density filter, where the density gradient is accompanied by a color gradient, normally based on the assumption that you want the sky to be seriously blue, and the ground to be seriously brown.
The effect is very very crappy, and I despise landscape photographers that go to this limits of tackyness.

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