One of the most important applications of liquid crystals is LCD (Liquid Crystal Display). In order to understand how an LCD works we should first take a look at the behaviour of a polarized beam of light passing through a liquid crystal material. Liquid crystals have the property to be birefringent. This simply means that the polarization status of the light gets split into two components, one is parallel to the direction (Ordinary component) of the director of the molecules and the other is perpendicular to it (Extraordinary component).
Components of light which are parallel to the director have a greater speed then the one perpendicular to it. The result of this effect is that when rays of light get back together on the way out after passing through the material, their polarization status will be changed.
This effect depends on the thickness of the material. Tuning the right thickness and the right orientation of the director we can basically have all desired states of polarization.

We know that if we send a beam of light through a couple of crossed polarizers, light doesn't make it through the second polarizer.

A beam of unpolarized light coming from the right passes through a pair of crossed polarizers. No light comes out of the apparatus.

Let's suppose to insert a liquid crystal between two crossed polarizers. After passing the first polarizer light gets linearly polarized along the direction of the trasmission axis of the polarizer. While crossing the liquid crystal the polarization status gets split in two directions having different speed as explained above. On the way out they join back giving rise to an elliptically polarized beam of light.
A this point the beam meets the second polarizer which will filter out all the parts of the beam whose direction in not parallel to its transmission axis.

A beam of unpolarized light coming from the right passes through a pair of crossed polarizers. The liquid Crystal (red box) is now in the middle. Linearly polarized light comes out of the apparatus on the left hand side of the picture.

Let's now suppose to orient the first polarizer in such a way that its transmission axis is paraller to the director of the molecules. In this case the polarization of the beam of light doesn't get affected by the liquid crystal and the second polarizer will stop it.

That was the basic mechanism behind an LCD. We created a system in which light can get abosorbed or can pass through two crossed polarizers (pixel off VS pixel on) dependently on the direction of the liquid crystal director.
If we find a way to modify the direction of the director as we want, we will have all the necessary elements for creating a liquid crystal display.

In order to address this second point let's consider the electric properties of the molecules of a liquid crystal. It turns out that they have a natural electric dipole, in other words, the two edges of each rod carry an opposite electrical charge. The consequence of it is that if we apply an intense enough external electric field to the liquid crystal the charges at the edge of the molecules will try to orient themselves along the direction of the inducting external field.

Dipoles try to align themselves along the direction of the inducting field.

We then just got the tool we were looking for.
We are now able to orientate the director of the Liquid crystal in any direction we like and we can then start thinking about how to make a display out of it.