Originally Posted by DCCommuter
The key to a bike steering is the "trail" of the steering, the fact that the contact point of the front tire is not directly in line with the axis of the handlebars. When you move the handlebars to the right, the contact point moves to the right. Moving the contact point to the right without moving the center of gravity causes you to lean left -- just like if you moved your feet right without moving your hips you would fall left. On a moving vehicle, falling to the side causes you to turn in that direction.
This is also how you can use steering to balance. If you start to fall one way, you steer in the direction of the fall. This moves your contact point under your center of gravity, and you stop falling. This effect is really noticeable at low speeds, such as going up a steep hill.
You have got it right in the second paragraph. The "trail" is the key to bike stability and not to steering. When you turn the bars to the right the contact patch will move to the right regardless of how much trail there is, because the wheel will roll in that direction. Since the contact patch is below the c of g this will cause the bike to lean to the left. A leaning wheel will turn in the direction of the lean. In a turn the trail will exert a force tending to centralize the bars, which gives the stability.