The longer the shutter is open, the more motion blur will occur. Occasionally motion blur is preferable, but more often than not, accidental motion blur ruins what could have been a great picture. A good rule of thumb is to shoot a motionless subject at or above 1/ to avoid a blurry picture. For example, if you're using a 17-55mm lens at 50mm, it's best to shoot at 1/50 of a second or higher to avoid blur. Everyone is different and some people are steadier than others, so if you're particularly steady you may be able to lower the shutter speed a bit, or if you're particularly shaky you may need to increase the shutter speed a bit. Shutter speed is most important when your subject is moving.
For example, when shooting a moving car, if the shutter speed is too fast, it will look like the car is standing still. Too slow, and now the car is a giant blur. 1/160 is my preferred setting to catch a little motion blur in the background and to see the wheels spinning, but everyone is different. A good example of how shutter speed can affect your picture can be seen below. This is a waterfall taken at 1, 1/3, 1/30, 1/200, and 1/800 of a second.
As you can see, at 1 second, the water looks silky and almost magical. At 1/800 of a second you just see individual water drops. At 1 second you can capture the full motion of the water, at 1/800 you can completely stop or freeze the motion of the water.
To recap:
A slow shutter speed will let in more light, but will increase the chance of motion blur.
A fast shutter speed will let in less light, but will decrease the chance of motion blur.
