How does an aircraft tailplane work?
For conventional designs, the tail is composed of two parts: the horizontal tail and the vertical tail. They play a role in the trim and the manoeuvrability of the aircraft but at different levels. The horizontal tail is mainly used for longitudinal stability (and trim) while the vertical tails used for the lateral stability (and trim).

About Stability

It is possible to talk of stability only after having defined an equilibrium point around which the stability is studied. An aircraft is in equilibrium if the forces and moments it experiences are balanced. Using a simple model for the longitudinal analysis, it can be decomposed in three relations called the trim equations. In order to keep it simple, it will be assumed here that the angle of attack and the flight path angle are zero. (Note that the same reasoning can be achieved with non-zero values but the equations then become quite messy.)

Longitudinal Equilibrium

These three equations are:

$$L=mg$$
$$T=D$$
$$M=0$$

where $L$ is the total lift, $mg$ is the weight of the aircraft, $T$ is the thrust, $D$ is the drag and $M$ is the pitching moment around the centre of gravity of the aircraft. The second equation won't be studied further since it does not help to understand the role of the horizontal tail and its influence. Looking at the following picture, one can see that usually, the centre of gravity and the point where the lift applies (called the aerodynamic centre) are not the same. This means that the lift generated by the wing creates an induced moment around the centre of gravity that one should add to the already intrinsic pitching moment due to the main wing (usually a pitch down moment for conventional airfoils).



Knowing that, it is possible to rewrite the two equations of interest including the contributions from the main wing and from the horizontal tail.

$$W+L_t=L_w$$
$$M_0+bL_t=aL_w$$

From these equations and the figure, it appears that the horizontal tail is used to generate a lift which induces a moment helping to balance the moments equilibrium and thus prevent the aircraft to spin on itself (pitchwise).

Drawback and Solution

From both the figure and the equations it turns out that the lift contribution from the tail is usually negative, meaning that more lift from the main wing is needed to keep a trimmed (or balanced) aircraft. This drawback can be overcome by the use of a canard configuration instead.

Lateral Stability

The same can be done for the lateral equilibrium and stability but there it is the vertical tail that is used. It is symmetrical so that there is no yaw induced and if there is some side force experienced, it will create a moment in order to reduce the side-slip angle.

Comparison of Lift Created by the Tail and Main Wing

For a trimmed configuration, it is easy to see that the lift created by the main wing is more or less the one created by the tail plus the total weight of the aircraft, which gives an idea of the difference between the two forces.