When a picture is taken in a fully automatic mode, the camera detects the amount of light using its light meter. Based on that, it calculates what aperture and shutter speed to use and takes the picture. This is fine, but in most cases, there is more than one combination that provides the same exposure. For example, an aperture of f/8 and a shutter speed of 1/30 produces the same exposure as f/5.6 at 1/60 shutter speed. However, this does not yield the same picture.
With a fast shutter speed, the camera stops or freezes the action. This is great for photographing things like plants outdoors (keeps the breeze from making your flower petals blurry). With a slower shutter speed, you can produce intentional blur. More common for the average photographer is the get a "soft" affect for moving water. With a fast shutter speed you can see individual drops (see Fountain Picture). With a slow shutter speed the water's motion causes those individual drops or lines to disappear and produces a more cloud-like affect (see Waterfall Picture). Many photographers prefer this look for things like fountains and waterfalls.
How do you get the picture you want? By using the shutter priority mode, the photographer gets to choose the shutter speed. The camera will automatically select the aperture to produce the proper exposure according to its light meter. If you want to stop the action you can set a fast shutter speed. If you want to allow for blur or softening you can set a slower shutter speed.