Consider a system with only the two energies 0 and \varepsilon. It looks simple enough, and yet a good many real systems are described by it: a spin in a magnetic field, an impurity in a crystal, a molecule with only two orientations. Each of them has two places to be.
The two horizontal bars in the figure are the levels. A particle is on one or the other, never between. The gap \varepsilon between them is the only measuring stick this system owns.
Once temperature enters, the thing to compare it against is kT. Whether kT is large or small beside \varepsilon: everything that happens in this system follows from that comparison alone.
The partition function is Z = 1 + e^{-\beta\varepsilon}, and the probability of the upper level is \dfrac{e^{-\beta\varepsilon}}{Z}.
The two curves in the figure are those probabilities. At low temperature the lower level is near 1 and the upper near 0. kT is not enough to lift anything to the top.
Raise the temperature and the two draw together, both heading for \dfrac{1}{2}. Once kT far exceeds \varepsilon the difference of \varepsilon might as well not exist, and the two levels are used equally.
The mean energy is \langle E \rangle = \dfrac{\varepsilon}{e^{\beta\varepsilon} + 1}. It rises with the temperature and then stops at \dfrac{\varepsilon}{2}.
The faint straight line in the figure is an ideal gas, put there for comparison. That one grows without limit in proportion to the temperature. This one cannot drive everything up to the higher level, and however hot it gets it goes no further than an even split.
The difference is whether there is anywhere above to go. A gas has as much room above as you like; this system has a ceiling at \varepsilon. Pour heat into a system with a ceiling and past a point it stops taking any in. That is what leads to the shape of the heat capacity in the next scene.
The heat capacity is the slope of the energy against temperature. The shape can be read off the previous curve: flat when cold, rising in the middle, flat again when hot. So the slope raises a peak somewhere between.
The figure is that peak. It is called the Schottky heat capacity, and its top falls where kT is about half of \varepsilon. It drops to 0 at low temperature because the system is frozen out, and to 0 at high temperature because there is nothing left to take in.
Two zeros of quite different origin stand on either side of one peak. Find a peak like this in an experiment and there are two levels there with a gap of \varepsilon. Measuring a heat capacity becomes a way of probing levels you cannot see.
The nearest two-level system to hand is a spin in a magnetic field. Along the field is the lower level, against it the higher, and the gap between goes as the strength of the field.
The imbalance of orientation is the magnetisation. Using the working of the previous scenes as it stands gives M = M_0 \tanh(\beta\mu B). The curve in the figure is that: it rises as the field strengthens and then levels off once all are aligned.
In a weak field the \tanh may be taken as a straight line, so the magnetisation goes as the field with a constant that goes inversely as the temperature. This is Curie's law. A law found by experiment has come out here too, from nothing but counting two levels.