In this photo nominated as Canon Australia Photo of the Week, the background seems to blur while the rider is sharp. Wouldn't this normally be the other way? How was this done?
This is achieved by what is known as panning.
To explain it simply, you basically follow your subject with your camera (in a panning motion.)
It takes quite a bit of practice to get it right, but patience and practice will pay off with great looking shots.
More information on capturing moving subjects can be found HERE.
Taking pictures of your children in action can be very good practice for this technique.
This picture was taken at close range with my mobile phone, so the quality is not the best, but just to train technique, you can use whatever at hand. Also it is not easy to get a clear focus on the target when he's running, as he's bouncing and wriggling.
It is easier to get clear focus with bicycle action...
For a good motion blur effect, you should set a slightly slower shutter speed than you normally would.
If you don't aim for the motion blur effect, you can use normal shutter speed, but follow the target in motion. This is not to intentionally get the background blurry for effect, but to make sure the target is not blurry. And it is good practice for the "follow target" technique as you would for a full motion blur effect.
Note in this picture the kid is not blurry, but the background has a slightly noticeable blur, although not to the extent of a full motion blur effect.
That effect was achieved following the subject with the camera as explained by the other answers.
There are, however, other popular methods used primarily in advertising - especially car photography, to enhance the feeling of motion and speed:
Mounting the camera to the subject will freeze the subject, but capture the motion in the background.
Photoshop: Separate/etch/select the subject, and selectively only use the motion blur filter on the background.
You can also achieve a great effect by zooming in or out at a lower shutterspeed.