Using fast shutter speeds can stop motion and prevent blurring but it can be great fun doing just the opposite. Using slow shutter speeds to intentionally blur moving subjects is one way to capture a creative image and convey motion whether real or an illusion.
Conveying a sense of people rushing or in a hurry is easily done by setting a shutter speed less than 1/15 of a second or more. Slowing it further can make them totally disappear. Try it some time. If you’re concerned that people are spoiling your scene, set a long enough exposure (3 or 4 seconds) and poof, they’re gone.
Busy places like airports and train stations are ideal for this type of photography. So are city streets during business hours, neighborhood parks on the weekends, Georgetown anytime of day, well, you get the idea. 
Panning is another way to imply speed and movement. Blur only the background and keep your subject sharp or blur them both. For a sharp subject, try 1/30th or 1/60th as a starting shutter speed. Lock focus on your subject and follow it thru the frame moving your camera along with them. The key is to move the camera parallel to your subject and to completely follow thru avoiding a sudden or jerky stop once the subject has passed.
Flowing water and waterfalls make beautiful compositions when the water is captured as a silky flow. 1/2 second to 2 or 3 full seconds usually works well enough. If your scene is very bright though, you may need a neutral density filter to darken it to avoid over exposure.
The list goes on and the possibilities to capture motion are endless; race cars, traffic in a city scene, car tail-lights at night, amusement park rides, children playing or sledding, bicyclists and more. 
There are even apps for the iPhone for this, my favorite is SlowShutterCam.
For best results when using slower shutter speeds, use a sturdy tripod (just check that you’re allowed to use it wherever you’re photographing) and a remote or cable release. Even though we’re deliberately trying to blur motion, hand holding at less than 1/60th of a second will result in the entire frame being blurred.
It’s also best to use a camera that offers the manual exposure mode for this type of shooting. Set a low ISO, set the shutter speed based on the speed that your subject is moving or the amount of blurring you want to achieve, then just set the aperture until you get the proper exposure reading on your indicator scale.
Learn the creative use of slower shutter speeds and add a new skill to your photography. Give it a try but watch out, it’s so much fun you may get hooked.
Shutter speed info from photos:
Jeep in Motion by Tim Cooper shutter speed 1/10th of a second
People at Tidal Basin in DC - 1/10th of a second
Bicyclist - 1/4th second
Train Moving - iPhone - SlowShutterCam
Photos by Marie Joabar unless noted.