Shutter speed is the length of time, normally expressed in fractions of a second. It is the period of time that the shutter is open for and has a direct effect on the nature of the picture.
There are loads of approximations, but generally speaking for sports you want to be over 1/250th of a second to freeze the action. The picture in your avatar looks like 1/500th+ as there is no hint of motion.
Night photography can see exposures go into the realms of minutes and even hours. Also, landscape and architectural photographers often use special sunglasses for the camera allowing a many hour exposure to get rid of people and parked cars.
Most walk around photography should be shot at at least 1/focal length to eliminate camera shake and blurriness when hand-holding. So if you've got a 600mm sports lens, you need a 1/600th of a second exposure or shorter. This is why people pay $$$$$ for lenses that are long and bright. Walking around with a 50mm f1.8 you can get away with 1/60th of a second easily, but 1/125th pretty much guarantees a reasonable result.