On the one hand, the shutter of a camera is used to set exposure by determining how long the sensor is exposed to light for any given shot. On the one hand, the shutter also determines how much movement is captured in an image. For images with fast moving objects you need a fast/short shutter speed so that the images don't appear blurred. In other situations you may slow/long shutter speed so that motion is apparent.
Terminology
Shutter speeds are measured in fractional seconds. However, when adjusting shutter speed cameras adjust them in increments to be related to fractions of a stop. For instance, if you were to increase the shutter speed from 1/50th of a second it would likely go to 1/60th of a second -- one third of a stop of light less. Moving again from 1/60th of a second it would go to 1/80th of a second -- another third of a stop.
Always, the faster the shutter speed, the less light available for the picture. The fastest shutter speeds on most cameras is 1/4000 of a second which is fast enough for almost any situation. Professional cameras typically go up to 1/8000. Shutter speeds can be as slow as you like though, even measured in minutes for late night shots.
Shutter speed is directly related to blur, which may come in 2 forms. First, there is motion blur from an object moving while the picture is being taken. Simply put, the faster the object is moving the faster the shutter speed needs to be. To my knowledge, there is no rule of thumb for a speed:motion blur equivalence. From my experience, you want about 1/50th or faster for walking/talking/party etc. For running or fast movement, 1/300th or faster.
The second blur is camera shake. The longer the focal length of the shot (the more zoomed in), the more camera shake is apparent. There is a rule of thumb in this case, but it relates to 35mm equivalent focal lengths. The rule is the shutter speed should be no shorter than one over the 35mm equivalent focal length to avoid camera shake blur. So, if you're shooting at 55mm that's 82mm equivalent you should be at 1/80th or faster or your image may be blurry even if there is no motion.
Vibration Reduction (aka image stabilization) is a way some lenses have of reducing camera shake. They're quite good, but not infallible. Typically, VR allows 1 to 2 stops slower shutter speeds without worrying about camera shake. So, at 55mm you may be able to shoot around 1/20th of a second without camera shake (although that's pushing it, honestly).
Shutter Speed
In order to manually set shutter speed, you'll need to change your camera's PASM dial to "S". The "S" stands for shutter priority. In other words, the camera will set everything for you based on the shutter speed you select. This is a great way to be able to control one aspect of your shot while the camera manages the rest.
I'll use an example of a fountain that has a very slow flow rate. Many fountains, streams, and waterfalls flow faster than this, but it will serve as a good comparison.
For our first statue shot, the shutter speed is set to 1/1000th of a second. The shot is well exposed and well composed (good enough at least), but the shutter speed was set so high that the fountain that typically looks like a stream of water now looks like droplets.
The second shot was reduced to 1/160th of a second (2 and 2/3rd stops longer than 1/1000). Again, at 1/160th the water distracts from the picture as it looks choppy.
At 1/80th (1 stop longer than 1/160) the water is beginning to look more natural. While there are still droplets, the water at least looks like it is flowing. Without lens VR or with people in the shot, this or 1/50th is about as slow as I would want to go. It would give a nice balance of water flowing but assurance that the shot will be blur free.
Now at 1/20th (2 stops longer than 1/80) individual droplets are no longer visible. However, you can still make out some break in each of the streams. While VR is preventing this shot from being blurred from camera shake, people would have to hold pretty still to not get any blur from their movements.
This shot is at 1/10th (1 stop longer than 1/20) of a second. VR and a steady hand have kept camera shake blur away, but I definitely would not have shot this slowly with people involved. The water now has a decided flow to it, and even the pool around the fountain is beginning to look less crisp.
Now down to 1/5th of a second (1 stop longer than 1/10th) any breaks in the streams are gone. The image is still relatively blur free, but if you zoom in very tightly you'll see the start of some blur from camera shake.
This shot was at 1/2 seconds -- a full stop and a third longer than 1/5 of a second. At this point the streams look fully blurred and even the pool around it seems blurred from the ripples of water. This clearly presents the image of a flowing fountain and moving water.
Here is the pool of water at 1/250th.
And again at 1/2. Notice how the foot of the statue is still relatively sharp (there is some camera shake if you zoom in), but even the reflections are now blurred.
In Use
Typically you will always want to use a shutter speed that is at least one over your focal length. That is a great rule of thumb for taking pictures of people and scenery. Nothing is more frustrating that the perfect shot that turns out blurry because the shutter speed was too slow.
When you get to a dimly lit room and don't want a flash, you may want to turn your PASM dial to "S" so that you can set a minimum shutter speed. Also, you may have a menu setting that allows you to set a minimum shutter speed that the camera won't go below even in automatic.
When you're trying to freeze action like a runner, always give yourself room if you can. If you're shooting at 200mm, 1/300th may be good, but if it's bright enough take 1/500th or 1/1000th if you can. Especially at very long focal lengths, very slight movements can introduce annoying camera shake.
If you're going around nature or water, take a tripod! That will allow you to set very low shutter speeds that can create surreal pictures, or just set the speed low enough that you can catch the motion of water without camera shake.
Trade-Offs
The faster the shutter speed you use, the less exposed the shot is. So, fast shutter speeds are easy on bright days, but indoors it's difficult to get a very fast shutter speed and still expose the shot. High ISO, large apertures, and flashes can help.
On the other hand, getting to slow shutter speeds may not be possible on bright days. The fountain shots above were only possible because it was a dark, overcast day. Lens filters can actually reduce light to allow for longer shutter speeds.
Technical Notes
If you use a tripod, TURN VR OFF. VR can over-correct if it's on a tripod and introduce shake.
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