An aeroplane must be in balance longitudinally in order to fly. This means that the net effect of all the forces acting on the aeroplane produces no overall pitching moment about the centre of gravity. Without a tailplane there would be only one combination of speed and centre of gravity position for which this requirement was met. The tailplane provides a balancing force to maintain equilibrium for different speeds and centre of gravity positions. Because the tailplane is located some distance from the centre of gravity, even the small amount of lift it produces can generate a large pitching moment at the centre of gravity.
An aeroplane with a wing only is normally unstable in pitch (longitudinal stability). This means that any disturbance (such as a gust) which raises the nose produces a nose-up pitching moment which tends to raise the nose further. With the same disturbance, the presence of a tailplane produces a restoring nose-down pitching moment which counteracts the natural instability of the wing and make the aircraft longitudinally stable. A stable aeroplane can be flown "hands-off" and will maintain the same altitude and pitch attitude.
A tailplane has a hinged flap called an elevator, which allows the pilot to control the amount of lift produced by the tailplane. This in turn causes a nose-up or nose-down pitching moment on the aircraft, which is used to control the aircraft in pitch.