The shutter goal is to allow light to hit the sensor or the film for a specific duration.  In SLR, the shutter takes the form of a curtain that opens.  A second curtain closes this opening when the elapsed time is over.  Some Cameras have an electronic shutter instead of a mechanical one.

To achieve very high-speed, the closing curtain will start closing the opening before the opening curtain has complete the full distance.

How about the shutter speed?

We refer to elapsed time were light reach the sensor or film as the shutter speed.  On older cameras, increment between one shutter speed and the next fastest shutter speed was one full stop.  One more stop of light is doubling the light.  This means that the next shutter speed below 1/250 second is 1/125 second and this effectively double the amount of light.  1/500 second would mean that only one half the light compared to 1/250 second.

Modern cameras can use ½ stop increment or 1/3 stop increment.  When turning the dial on a camera configured for ½ stop increment, 1 full stop needs two clicks.

Which shutter speed should I use?

As a rule of thumbs, the recommended shutter speed to avoid blurry pictures caused by camera shake is 1/[focal length].  Should you have a 50mm lens on your camera, a shutter speed of 1/50 second would be a good basis to avoid camera shake.  All people being different try it out and take good note of what you can achieve.

Very fast shutter speed are ideals for actions scenes where the subject is moving quickly, conversely, slow shutter speed might be needed when there is less light or simply because we want to produce a blurry picture.  There are multiples reasons to use a slower speed.  In the example above I wanted the water to be blurry and I intentionally chose a slower speed.  In order to avoid camera shake, I used a sturdy tripod.