Is the fine structure constant actually a constant or does its value depend on the energy scale? The value of the fine structure constant is given as
$$
\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} = \frac{1}{137.035\,999..}
$$
It's value is only dependent on physical constants (the elementary charge $e$, speed of light $c$, Plancks constant $\hbar$), the vacuum permitivvity $\varepsilon_0$) and the mathematical constant $\pi$, which are considered to be constant under all cirumstances.
However the Wikipedia article Coupling constant states

In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127.

I don't understand how this can be possible, except that one of the physical constants above or even $\pi$ are actually not constant, but dependent on the energy scale. But that seems nonsense. 
So what do physicists mean when they say that the fine structure constant $\alpha$ increases with energy? Can you perhaps reformulate the quoted sentence above so that it makes more sense?
 A: Expanding on what Vladimir said: the thing that is changing with energy is $e$ (the others are not constants so much as conversion factors between length and time, time and energy, etc.). The reason the charge can vary is that the vacuum is not entirely empty. Sloppily speaking, near a charge, the electric field interacts with virtual (electron/positron) pairs and the effect is that the virtual pairs screens the "raw" electric field. Thus, if you're far away, you see one value, but as you get closer the electric field raises faster than $1/r^2$. With scattering experiments, how close you get to a charge is directly related to the in-going energy of the particles. Now, in modern physics, we account for this by saying that the charge $e$ changes with energy scale; this sounds bizarre in the form I just explained (since you might expect that we just declare the force to be not $1/r^2$), but it turns out that this is the neatest way intellectually to understand it, due to a convergence of issues to do with wanting to preserve observed symmetries in the theory at all scales.
Incidentally, for things like colour charges in QCD, the vacuum anti-screens, which is to say that the observed field strength increases as you get further away. Heuristically, this is what leads to confinement of quarks in the "normal" phase. 
A: Do not worry, it is a constant. 
There is a sloppiness and misunderstanding in physics when the energy dependence of the cross sections is wrongly attributed to the "fundamental constant" whereas it is a cross section feature.
