Long exposure photography, as the name implies, simply refers to photographs taken with relatively long exposure times. Typically, the photographer wishes to capture the movement of one or more elements within a scene over the course of the exposure. The resulting photograph then reflects this movement in some way. This technique can be used to create various effects, and thus is a useful technique for photographers to know.
Taking long exposure photographs can be a bit tricky. First and foremost, the photographer needs to imagine what is going to happen in the scene over time and how that will affect the final result. There are several technical aspects to take into account as well.
As with all forms of photography, managing the light in a photograph is important. Long exposure times can add yet another layer of complexity to lighting. Not only does the light build up over a longer period of time, but it may also move around and/or change in intensity. The exposure settings on the camera also govern how the movement will be captured, including how much overall movement is captured and how much each element of the scene contributes to the final image. Many factors, such as how bright an object is, how reflective it is (e.g. glossy vs matte), how fast it is moving, the use of flashes, and so on, all play a role in how an object will show up in a long exposure photograph.
Secondary effects, such as camera shake and sensor noise, have a higher chance of affecting long exposure photographs as well. Digital camera sensors tend to generate more noise when they are exposed for longer periods of time. Camera shake can become a factor as well. Even small, subtle movements can affect the sharpness of the image. For example, cameras are commonly placed near roadways when photographing the light trails created by cars driving at night. Vibrations caused by those cars can subtly and constantly shake the camera, introducing a slight blur in the final photograph.
More examples of long exposure photographs that I have taken can be found here.