It's common for people to feel nervous about falling off a treadmill, so they change their running form and don't use the same running form as they use outside. You should be running on the treadmill the same way you would run outdoors. Try to run with your natural gait, and avoid taking short, choppy strides.
Another common form mistake is overstriding, or landing heel first with your foot well ahead of your body's center of gravity. Since the treadmill's belt is moving you forward, overstriding creates a braking force with the belt. To avoid this, try to keep your feet under your body, not ahead or behind it.
More: Proper Running Form
Video: How to Run on a Treadmill