Considering electronic band structure in solid state Physics, the Fermi level is defined as the chemical potential appearing at the Fermi-Dirac distribution
In that sense, it is just one specific value of energy that we chose to give a name to it.
Now, the Fermi level is quite important because of the following points, according to Wikipedia's article:
In an insulator, $\mu$ lies within a large band gap, far away from any states that are able to carry current.
In a metal, semimetal or degenerate semiconductor, $\mu$ lies within a delocalized band. A large number of states nearby $\mu$ are thermally active and readily carry current.
In an intrinsic or lightly doped semiconductor, $\mu$ is close enough to a band edge that there are a dilute number of thermally excited carriers residing near that band edge.
In other words, it seems that the conductivity properties of the material are determined by $\mu$.
But why is that? Why $\mu$ has all these properties?
How can we actually find out these properties of the Fermi level? How can we find out that the Fermi level determines the conductivity according to these points?