Neutral density (ND) filters come in various flavours but in reality a budding photographer needs to know very few things when using them without going heavily into the science of things. First thing to know is what a ND filter does.
ND filters modify the wavelenght of light in either reducing it or changing it. What that means in practical terms is that you can use them to change two things: aperture and exposure time. This in turn allows you to control motion blur.
The most widely used effect for ND filters is to create a motion blur of flowing water. You might have seen them used in waterfalls where a nice sharp scene of the environment around the waterfall is contrasted by the deliberate blur of water falling down. Water is a popular theme. Another way of use of ND filters, and one you might have seen, is the water like sand feature induced by ND filters of the sea. These can be achieved with certain filters even in the middle of the day.
Speaking of middle of the day, using an ND filter when very bright light is available can allow you to shoot at a wider aperture even when you are at your fastest shutter speed. This is very useful if you still want to isolate subjects and so on.
A ND filter less known but used also by photographers is the gradient ND filter. These filters provide the full effect of reducing light on one edge and gradually become less pronounced as you look through the other edge. While they can be found in round filters a lot if professionals use the square type filters. The reason they are more popular in those is that they offer flexibility.
Imagine you want to shoot a landscape during a dark cloudy day. The problem in these sort of situations is the camera light compensation feature that is so handy during bright days but so frustrating during cloudy ones. Cameras try to balance the scene you frame by averaging the light that is available. However, during a cloudy day the little light that comes through is from the sky. You have then two options. Either shoot two pictures at different exposures where one has the ground well lit and the other the sky and then merge them in photoshop (sounds like a lot of work) or you can use a gradient ND where you put the ND edge to the sky and thereby balance the light available.
Square filters are practical in the sense that you can quickly slip one in front of your lens and even add another one on top if you require more control of the light. Screw on filters require a little more time to put on and can have the problem of rotating the gradient out of position accidentally. However, they work the same way.
Finally one thing one has to keep in mind is the type of ND filter. These filters come in different strengths and usually this is indicated by a number (e.g. ND2, ND8 or ND 0.3, ND 0.6 etc...). These numbers usually indicate how many stops they reduce the light or the optical density of the filter. So an ND2 filter (lowest optical density ND 0.3) will reduce light entering the sensor by 1 stop. The next one up (ND4 or ND 0.6) will reduce by 2 stops and so on. However different manufacturers might have different ways of labeling their filters so check with your own camera to see how many stops the filter is actually reducing the light by.