Usually, you would want to avoid this effect by using shutter speeds of 1/125 or faster. Certain subjects, however, are most effectively photographed by deliberately capturing the blur.
Sports photographers frequently make use of blur to emphasize the speed or direction of movement.
There are two popular methods for doing this.
Subject blur - hold the camera steady while the subject moves
Background blur - move (pan or tilt) the camera with the subject.
People in action generally require a slower than average shutter speed (1/50th of a second or longer.) High speed subjects such as a race car or a roadrunner moving across the camera view, may blur at 1/500.
If possible, you will want to bracket your shots, because the subject may move faster or slower than you anticipate.
When adjusting to a slower shutter speed, you will also be allowing more light to reach the sensor/film for a longer period of time. In order to avoid overexposure, you will need to use a smaller aperture (larger f-stop number).