What "real life" problems can be solved using billiards?
I am very surprised that nobody mentioned the REAL reason why mathematicians are so much interested in billiards. It comes from mechanics (celestial mechanics, first of all), and the first question considered was that of existence of periodic trajectories. Which correspond to periodic orbits in mechanics.
G. Birkhoff in "Dynamical systems" explains: "Before we give an example illustrating applications of Poincare theorem and its generalizations, let us consider first of all a special but very typical problem of this sort, namely the problem of motion of a billiard ball on a table which is bounded by a convex curve. This system is of a great interest for the following reasons. Every Lagrangian system with two degrees of freedom, ... etc."
For an easily available source, see Birkhoff, On the periodic motions of dynamical systems, Acta math., 48, 1927.
There are some potential applications to the design of optical cavities for lasers.
Imagine a region whose sides are mirrored. You might shine a laser in, and have an opening or a partial reflector where the light can come out. Typically, there is some "gain region" in the interior, say a crystal which is excited electrically or by a laser operating at another frequency. We want some input beam to pass through the gain region many times, or to spend a lot of time there on average, before the beam hits the output.
A complication not present in the usual mathematical billiards is that the gain region may have a different index of refraction from the surrounding medium, so light entering it at an angle may be deflected. This can even depend on the intensity of the light.
One possibility is to control the geometry very precisely. In fact, this is one place you can use the fact that hyperboloids of one sheet are doubly ruled surfaces: You can make the even segments follow one ruling while the odd segments follow the other ruling. However, aligning this precisely can be tricky, particularly with the complications above, and sometimes it doesn't make efficient use of the volume, so you don't get as many passes through the crystal for the space you allocate to the cavity.
You might want to design a cavity so that it tolerates small errors, so many paths spend a long time in the gain region and then exit.
The billiard-ball computer, also known as a conservative logic circuit, is an idealized model of a reversible mechanical computer based on Newtonian dynamics, proposed in 1982 by Edward Fredkin and Tommaso Toffoli. Instead of using electronic signals like a conventional computer, it relies on the motion of spherical billiard balls in a friction-free environment made of buffers against which the balls bounce perfectly. It was devised to investigate the relation between computation and reversible processes in physics.
The billiard-ball computer was never realized in this form, but it played a significant role in the development of the quantum computer. Since the unitary evolution of quantum mechanics is reversible, it cannot employ the irreversible logical operations of a conventional computer. (This story is told here.)
For an altogether different application of billiard ball dynamics, to semiconductor device physics, see Billiard model of a ballistic multiprobe conductor (1989).