Depending on the ND's opacity, they can be used to avoid blowing out highlights when the camera's aperture cannot restrict far enough, or when you don't want to use a minimum aperture to avoid diffraction and maintain detail.
They are also very handy for IR photography - if you get an IR cut filter, often it can be useful to stack an ND8 or even two (ND4, ND8) filters to get deeper blacks in skies and water.
And NDs are invaluable when you want to do slow-shutter work in daylight...such as photographing waterfalls with ribboned water effects, and blurring people into ghost streaks as they walk around. Most cameras won't be able to close down the aperture enough to allow several-second shutter speeds in daylight. Figure for around 2 stops of light restriction with an ND4 (.6), and three stops for an ND8 (.9) filter. Then there are the serious ND filters, such as my ND400, which has approximately a 10-stop restriction. This allows you to fire off 30-second shutters in the middle of the sunny afternoon.
These are all solid ND filters. As mentioned above, the graduateds are useful for darkening parts of the photo more than others.
Polarizers do usually offer some light restriction - usually around 1-stop loss. But primarily, you can reduce glare and reflection off of reflective surfaces. It's particularly fun to stand in front of a plate-glass window which is reflecting the scene behind you, look through your polarizing filter so you can see all the reflections, then slowly turn it 90 degrees and watch the reflections disappear and the interior behind the glass suddenly coming into view.