Shutter Speed

Shutter speed is your best friend and your worst enemy in photography. If you get it right, it can really make an image perfect, and if you get it wrong, it can make a great image terrible. Shutter speed is how long your cameras shutters are open (these are what expose the digital sensor or film to light). This of course is dependant on the light conditions of the scene.

What if the shutter is open too long?

Simply put, you get blur. This could be what your intending, or this could be a mistake, but you must be aware that having the shutter open for a while results in blur. You can change this by using ISO, or by using Aperture, which will speed up the light intake to the sensor, and reduce the amount of time that the shutter is left open.

Can the shutter close to quickly?

In a manner of speaking, no. Your shutter on most cameras will select the appropriate time to be open, but it can shut too quickly if you are setting it manually, and thus not let enough light in. Shutter can close too quickly also when using flash, but cameras do tend to compensate for this, and compacts/bridge cameras don't have this problem.

The uses of slow shutter speed

Slow shutter speed will get blur, and this can be very useful indeed in the right circumstances.
Slow shutter speed can be used for:

Slow shutter on a compact. This can be achieved by turning the ISO down as much as possible. This is the only way to do this on a compact.

Fast shutter is also important, and will be used more often.
Uses of fast shutter.