How does the Particle in a box model help research/understanding of physics? With my complete non-formal education on physics (meaning I read things around) I am having trouble to understand how the particle in a box model helps the further understanding of physics.
Didactically speaking, it is clear that it helps the understanding of how particles behave differently than macroscopic objects, but does it go further? Is the particle in a box model helpful for more than teaching students how physics work? How? Where?
 A: Absolutely! The particle in a box is the very first example most people see of a bound state problem. These are a class of quantum mechanical problems whereby we see, by simple mathematics, that the energy levels of certain quantum systems are discretely quantized. While this doesn't sound particularly exciting (because much cooler problems like the Harmonic Oscillator and the Hydrogen Atom exist and exhibit the same phenomenon), it is still a neat little bit of physics.
So how does this help in the research world? Short answer: it doesn't. Long answer: it does. A lot.
Let me explain.
While the actual result of the energy level spacing and exact quantization procedure used in solving this model don't really do much for you in modern research, there are a couple of very nice features about the Particle in a Box that are used everywhere (especially in Condensed Matter, Statistical Physics, and Quantum Field Theory). The idea is simple: sometimes when trying to solve a problem that contains a free particle (a particle whose Hamiltonian is of the form $H=p^2/2m$), it becomes convenient (and almost necessary) to consider the particle being confined to a volume $V\equiv L^3$ of a box of side lengths $L$ and demand that the wavefunction is periodic at the boundaries.
How does this help? Well, just as in the particle in a box, the introduction of the finite volume of space tells us that our momenta can no longer take arbitrary values, but must, instead, be quantized as
$$\textbf{p}=\frac{2\pi\hbar}{L}\textbf{n}$$
Where $\textbf{n}=\langle n_x,n_y,n_z\rangle$ is a vector of integers. From this, we easily see that the energy levels are of the form $E_{\textbf{n}}=2\pi^2\hbar^2(n_x^2+n_y^2+n_z^2)/mL^2$. This seems just like the Particle in a Box energy with an extra factor of four (this factor comes from the fact that we chose periodic as opposed to vanishing boundary conditions at the edge). This doesn't seem particularly useful until we start trying to calculate sums over momenta. Using this discretization, however, one can show the correspondance
$$\sum_{\textbf{p}}f(\textbf{p})\sim V\int\frac{\mathrm{d}^3\textbf{p}}{(2\pi\hbar)^3}f(\textbf{p})$$
You might still be skeptical. "When will this actually be used?" you might ask. So let's do an example.
This example requires some basic knowledge of Statistical Physics, but shouldn't be too hard to understand otherwise. Recall that we define the partition function of a theory at temperature $T=1/k_B\beta$ as
$$Z=\sum_{\alpha}e^{-\beta E_{\alpha}}$$
Where $\alpha$ runs over all indices that are used to quantify the energy. From this, we can calculate the entropy as
$$S=k_B\frac{\partial(T\log{Z})}{\partial T}=k_B\log{Z}+k_BT\frac{\partial\log{Z}}{\partial T}$$
(Don't worry if you haven't seen this -- it's the moral of the story that counts.) Let's calculate the entropy of a gass of $N$ free particles in some volume $V$. The indices of our theory are $\textbf{p}$, and so the single-particle partition function is
$$Z=\sum_{\textbf{p}}\exp\left(-\frac{\beta\textbf{p}^2}{2m}\right)\sim V\int\frac{\mathrm{d}^3\textbf{p}}{(2\pi\hbar)^3}\exp\left(-\frac{\beta\textbf{p}^2}{2m}\right)=\frac{4\pi V}{(2\pi\hbar)^3}\int_{0}^{\infty}\mathrm{d}p\,p^2e^{-\beta p^2/2m}$$
This is a simple Gaussian integral and can be readily computed as
$$Z=V\left(\frac{mk_BT}{2\pi\hbar^2}\right)^{3/2}$$
The $N$-particle partition function is $Z_N=Z^N/N!$ (assuming indistinguishable particles), and so
$$\log{Z_N}=N\left(\log{V}+\frac{3}{2}\log\left(\frac{mk_BT}{2\pi\hbar^2}\right)\right)-\log{N!}$$
And now we can compute the entropy of this gas of $N$ non-interacting particles as
$$S=Nk_B\left(\log{V}+\frac{3}{2}\log\left(\frac{mk_BT}{2\pi\hbar^2}\right)\right)-k_B\log{N!}+\frac{3Nk_B}{2}$$
Using Sterling's approximation, $\log{N!}\sim N\log{N}-N$, we arrive at
$$S=Nk_B\left(\log{V}+\frac{3}{2}\log\left(\frac{mk_BT}{2\pi\hbar^2}\right)-\log{N}+\frac{5}{2}\right)$$
This, which we would not have been able to derive without explicit knowledge of the Particle in a Box problem is the famous Sackur-Tetrode equation.
While this example seems very niche (and that's because it kind of is), the insights derived from the Particle in a Box problem are very widely used. In my personal research, I use it at least once a day (if not many many more times).
EDIT: Fixed a small typo in the summation-to-integral rule.
A: The particle in a box with infinitely high wall is very useful.
First, it shows that quantization of energy is qualitatively related to standing waves.  Because it is easy to solve, it is also a useful crude approximation for several bound state problems.  
The Cartesian 2D or 3D versions exhibit degeneracy, which can be used to understand the appearance of "shells".  The spherical version, for instance, can be used as an entry point to the ordering of nuclear energy levels. 

With a bump in the centre, it can be used to qualitatively understand the inversion motion of nitrogen in the ammonia molecule.  
Many-body wavefunctions are easy to combine so it is easy to explore some aspects of many body systems such as the exclusion principle.  This model (particularly the spherical 3d one) can be used to derive density of states $dN(E)$ as a function of the energy, or $dN(p)$ as a function of the momentum.  Because of the infinite discontinuity at the wall, there are some mathematical subtleties which can be illustrated using this very common example.
More interesting perhaps is the box with walls of a finite height (or a well with finite depth).  Although the solutions require a bit more sophistication (solving transcendental equations), the finite height of the wall makes it more realistic.  For instance, the Wood-Saxon potential of nuclear physics can be approximated by a particle in a box of finite height.
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A pair of separated wells of finite depth can be used as a crude model of a singly-ionized helium atom.  It is also a crude model for quantum dots.  
The list goes on: well problems are conceptually rich because they are familiar, manageable, and the mathematics is familiar.
A: Many systems that have practical applications use particle in a box theory. For example, the basics of semiconductors are derived using the potential well model, in this case the theory developed is called Kronig Penny model. Please read about it.
Also before solving complicated systems like large molecules and their energy levels, it's always better to start by the simplest system of particle in a box and develop intuition for quantum mechanics.
