After you have been running for a while, your running form will improve as you condition your body. Don't worry too much about form. One of the greatest runners of all time--Emil Zatopek--looked much worse as a runner than you will. Emil thrashed and grimaced. His head bobbed and his arms thrashed. He also won four Olympic gold medals.
Your form is largely dictated by your biomechanics: how your feet hit the ground. If you swing one arm wildly, it may be for natural reasons to balance what your legs are doing. Nevertheless, some basic statements can be made about "good" running form:
BODY: Run upright. Your back should be straight, roughly at a 90-degree angle to the ground. Ignore anybody who tells you to "lean into it," even when running uphill.
HEAD: Look straight ahead. Unless you are enjoying the scenery, your eyes should be focused straight down the road on a point moving about 10 meters in front of you. Try to run in a straight line.
ARMS: Swing your arms naturally. The angle at the elbow between your upper and lower arms should be about 90 degrees. Your hands should be loosely cupped, about belly level. Let your arms swing in rhythm with your legs. The legs should control armswing, not the other way around.
FOOTPLANT: The most natural landing is mid-foot, the ball of the foot landing first, the heel contacting the ground a fraction of a second later. The toes push off a fraction after that. Some runners land further forward, or backward, than others, based on what feels natural to them. Attempt to modify this natural gait at your own risk.
Your first steps may be awkward. After you have been running for a while, your running form will begin to improve somewhat as you condition your body. A good coach may be able to suggest some form improvements, but most runners adapt the form best suited for them without much prompting.
Form, By Hal Higdon. Taken from Runner's World.