Shutter Speed is the amount of time that the shutter of the camera is open. Depending how long you set the shutter speed for it could vary from letting lots of light in, to very little. Most of the time a person will use a shut speed of 1/60th of a second or faster. I went experimenting with my camera tonight to see if I could figure out what this meant exactly, and I seemed to have some luck!
The slower your shutter speed is, the more light it will let in, which means the brighter the picture is. The faster your shutter speed is, the less light it will let in, which means the darker the picture is. If you were to take a picture in the daylight and put the camera on the slowest shutter speed the picture that will appear will be completely white!
What shutter speed do you want to use to get motion in your shot? The slower the shutter speed is, the more motion will be shown in your picture. If you want to shoot an action picture, but not show the motion, you would use a faster shutter speed.
The images below show the pictures I took tonight when I was playing around with the shutter speed. They go from slow shutter speeds to fast shutter speeds. More detailed on shutter speeds can also be found here.