Say you have a toddler in a big swingset and you want to keep the baby in focus but have the background look like it's rushing by...
Or take the following example that I've shown you before. See how the motorcycle looks like its in motion? You can almost hear the rev of the engine...
How do we do this? How do we capture the excitement and motion of a moving subject in pictures? PANNING. It's a simple technique but it may take you a few tries.
Essentially all you do is follow the subject with your camera. Keep the subject in the same part of the frame so they stay as steady in your picture as possible. Say it's your child in a race, and he is running from left to right toward the finish line. Start on the left, and with a smooth and steady motion (move your whole upper body, not just your hands, think "sweeping") follow him as he moves right, snapping the picture as you go... at just the right moment!
Because you are moving the camera, the background is changing in your photo, which is why is becomes blurred in the direction its moving. Your subject on the other hand should, as I said, remain fixed within the frame (the frame is the edges of the photo, like a border).
On your first few tries, you might end up with a blurry subject. I recommend trying this out a few times, and taking your shot more than once if possible. It's fun and adds an exciting element to your pictures.
Go nuts... and keep those pictures coming in for the contest! ONE WEEK LEFT! :-)