Now that we know when it may be advantageous to use slow shutters speeds for photography, I will give you a basic run down on how to do this with your digital camera. Before even setting a single setting on your camera the first thing you need to do is determine approximately what length of exposure you are going to need as some require much longer exposures than others. For those images made to show speed you will want a speed slower than 1/60th of a second but fast enough to leave enough of the image in clarity. 1/30th of a second is a good place to start and you can adjust from there. Moving water images work very well with approximately 1 second or longer if possible. For those in the second category you will have to play around with exposures everywhere from 5 seconds to 15 or more minutes.

click to enlarge
In order to take a photograph with one of these slow shutter speeds you want to use the shutter priority or manual mode on your camera. Shutter priority mode allows you to dial in the shutter speed that you want to use and it will choose the necessary aperture to make a proper exposure. This method works well for those images depicting motion. For images involving the motion of water, aperture priority mode may be a better choice. Since water motion requires exposures of around 1 second, try stopping down the aperture all the way (highest F number possible) thereby giving the longest exposure possible with a proper exposure.
With night photography and those in the second category, manual mode is the way to go. Set up up the camera on a tripod, choose an aperture (to maximize sharpness or to maximize light), and, depending on the subject, choose an exposure time or BULB mode. In order to minimize camera shake for all of these in the second category, I recommend that you either use a shutter release or the two second timer.
For those point and shoot cameras without a manual option you can still get slow shutter speeds by knowing what the computer in the camera is doing during its
scene modes. For slow shutter speeds try night mode or fireworks mode.