Shutter speed plays a huge role in how motion is portrayed in a photograph.
If the shutter speed is high enough, moving objects will appear to be frozen. This is the case in the image below. The shutter speed was approximately 1/320th of a second, almost fast enough to make the water droplet appear totally frozen. As it is, the water drop appears mostly frozen with just a little blur around the edges. Not a perfect example, but you get the idea.
In the next image, I've slowed the shutter down to one full second. The water drops have now blurred together, looking like a trail of dust or a strand of spider web. Had I turned the water up a little higher, it would be more visible, but I want to compare oranges-to-oranges here and only change the shutter speed. The thing to remember here is that a slow shutter speed blurs moving objects.
Like I said above, a slow shutter speed causes moving objects to look blurry. However, this rule applies to anything that moves, including the camera itself! If the camera moves when you're using a slow shutter speed, everything in the image becomes blurred. This is called "camera shake".
In the top image, the fast shutter speed is "freezing" all motion, whether we're talking about the falling water droplet or my unsteady hands. In the image directly above, a tripod is holding the camera still, so the slow shutter speed isn't causing any camera shake. Only the water is moving, therefore that is the only part that appears blurred. In the bottom image, I am hand holding the camera while using a one second long exposure. Blur city. You can see why camera shake is considered a bad thing...
So how fast does your shutter speed need to be to prevent camera shake? The answer is about 1 to 1.5 times your focal length. We'll cover focal length later, but just file that away in the back of your mind for now.
Ok, so what did we learn? Use a fast shutter speed to freeze action and keep your images free of the dreaded camera shake. Use a slow shutter speed (and a tripod) to convey motion, whether you are shooting traffic or waterfalls. Handhold at a low shutter speed if you are forced to in dark conditions, but be sure to hold it steady as a rock if you don't want a unrecognizable blur instead of a photo.
Up next... ISO.