Technically, you should not be able to show motion in a still photograph. After all, the image on the paper is not moving; it's not going anywhere. But your mind takes the blurred image and tries to make sense out of it:
Understand the basic concept. Panning works when you move the camera in perfect motion with the subject. It's not enough to just swing the camera from side to side. You have to move it in perfect synch with your subject. Choose the right subject. Generally (and up to a point) it is easier to pan with a fast-moving subject than a slow one. Sprinters running sideways to you are great examples. They are moving fast enough that you can pan smoothly with their motion, and they are running in a straight line. People walking are almost impossible; they are too slow to get much blur and it's difficult to pan smoothly. Football players are tough because they move erratically.
Use Manual Exposure or maybe Shutter Priority metering. Whichever you choose, the object is the same. You don't want the shutter speed to change while you are shooting.
Pick a good shutter speed. This is important; however, there is no "correct" shutter speed for panning. The longer the shutter speed, the more blurred the background will be. A long shutter speed will make your subject pop out from the background, and that is good. But the longer the shutter speed, the more difficult it is to get the subject reasonably sharp. It's a balancing act. As a starting point, let's go back to the example of the sprinters running across the picture. Try anything between 1/8 and 1/60 of a second. Beyond 1/8 of a second it's really tough to get sharp, but it can be very interesting. Above 1/60 of a second, the camera will probably stop too much action and ruin the effect. Except for low-flying jets at air shows. Then you might need 1/500 second, and that brings us to our next problem.
Find the right background. The right background is almost as important as the right subject. The background must have some detail in order to produce the pleasing streaks you are looking for. That is why the jet is a bad subject for panning when it is up against a plain blue sky. Pan all you want but the sky will still be a featureless blue. Nothing will look as if it "moved." On the other hand, backgrounds with too much contrast will often make bad backgrounds for panning. Just one person in a white T-shirt can create an unsightly white blob in your photograph. Choose carefully.
Use the viewfinder correctly. Your viewfinder is your friend when it comes to panning. The best trick is to find a focusing mark in your viewfinder and put it on your moving subject. Now, try to keep that point perfectly aligned with your subject. Crosshairs would be perfect, but we don't have them in camera viewfinders, so we have to make do with what we've got.
Practice panning smoothly. Fluid, smooth motion is the name of the game. No jerking, no rushing, no hesitation. Stand with your body facing where you ideally want to shoot the picture, then rotate your shoulders to pick up your subject in the viewfinder. Start shooting before your subjects reach the ideal point; keep shooting after they pass that point. Follow through just like a good golfer. And practice. Good panning shooters literally go out and just practice their movements.
Go for the Goldilocks Effect. The combination of subject motion, panning, and shutter speed is not a precise science. Don't be afraid to adjust to conditions.
Try. Evaluate. Retry. Experiment! There is no right way, just infinite variables that can produce interesting results. For instance, if instead of "panning" you could rotate your camera at the same speed as the turning of the carnival Ferris wheel, you might get something cool.
Adapted from Panning 101 by Jim Richardson, http://photography.nationalgeographic.com/photography/photo-tips/motion-photography-panning-richardson/