This is my favorite question.

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.

Typically a cyclist uses lean to turn and uses the handlebars to fine tune his position.

A bike without trail would be impossible to balance or steer.