Neutral Density or "ND" filters come in 2 types: graduated, where they fade from clear to gray, or normal which are all gray. This article will be focused on the more popular completely gray ones, but once you understand the concept it's not difficult to figure out how to use the graduated ones as well.
As I said before, ND filters are completely gray and they come in different strengths. Many manufacturers label their filters 0.3, 0.6, and 0.9 while others may label them 1X, 2X, and 3X. Either way, they block 1 "stop" of light, 2 "stops" of light, or 3 "stops" of light respectively. -More on that later.
If you take a series of photos, each with a different strength filter, then one more photo without an ND filter at all, you might be surprised to find out that none of the photos would look any different from the others. Think about it this way, your camera can adjust for shade or indoor lighting vs bright sunlight, right? Well, it adjusts for the darkness that results from the filter in the exact same way.
So why would you use an ND filter then?
ND filters cut the amount of light that enters your lens. This may not effect the look of your photo, but it does effect the settings for shutter speed, aperture, and ISO that the photo is taken with. Let me give you an example of when you'd use an ND filter:
Let's say it's really bright outside and you are shooting a waterfall. You want to blur the motion of the water to get a milky effect like this. So, you set your camera to the lowest ISO you can, which we'll say is 100. And you set your aperture to f/16 (a sharp, high-number to cut down on the amount of light in your photo), and the shutter speed that you have to use to get the photo properly exposed is 1/45. -That's likely to be too high to blur the water much at all! ...so you disregard the sharpness loss of setting your aperture to f/22 which eliminates 1 stop of light. That will give you a shutter speed of 1/30 -still not enough to blur the water and (as is the case with many lenses) that's the highest aperture you can set with the lens you are using, so there's nothing else you can do via settings to get a lower shutter speed to blur the waterfall. ...but when an ND filter will help you!
* A 0.3 or "1X" filter will eliminate 1 stop of light which would give you a shutter speed of 1/22.
* A 0.6 or "2X" filter will eliminate 2 stops of light. ...that'd get you a shutter speed of 1/15. Now we're getting closer...
* Your 0.9 or "3X" filter will eliminate 3 stops of light. ...that'd get you a shutter speed of 1/11. ...that would do it if it's a fast-flowing waterfall.
I find ND filters that remove 3 stops of light or so to be the most useful. Also note that if you have a set of filters you can stack them to get an additive effect.
* If you stack the 0.3 and the 0.9 that'd remove 4 stops of light which would give you a shutter speed of 1/8 which usually blurs waterfalls very well.
* Stacking all 3 would reduce the light by 6 stops which would give you 1/4.
* ...but note that stacking creates more glass to shoot through, likely lowering your image quality. High-quality filters may be okay, though, especially if you don't crop the photo or print it at it's maximum size.