The definition of skin depth is mainly applied to conductive media and, as you stated, is the distance that the wave must travel inside a medium in order to experience a decay by $ 1/e $. This decay is due to the medium to be lossy.
The fact that this medium is lossy implies that the propagation constant of the wave becomes complex and this loss is related to the value of the conductivity. The process to obtain the relation is explained at Chapter 1 in the book "Microwave Engineering" by <PERSON>.:
First, we get the wave equation, or <PERSON> equation, from <PERSON>'s equations assuming an harmonic (cosine and sine dependence) time dependence:
<PERSON>'s Law of Induction: $\nabla$x $ \vec{E} = -j\omega\mu\vec{H} \space\space\space\space\space\space\space\space\space(Eq.1)$
<PERSON>'s Law: $\nabla$x $ \vec{H} = j\omega\epsilon\vec{E} +\sigma\vec{E}$ $\space\space\space\space\space\space\space\space\space(Eq.2)$
being this last term corresponding to $\vec{J}=\sigma\vec{E}$, the surface current density.
Here, j is the imaginary number $\sqrt{-1}$ and $\omega$ corresponds to the angular frequency $(2\pi f)$. $\mu$ and $\epsilon$ correspond to the magnetic permeabilityand dielectric permitivity of the media.
These equations, in their most basic and simple meaning, state that a variation in time of a magnetic field generates an electric field and that a variation in time of an electric field and/or the presence of an electric current surface generate a magnetic field.
By applying some vector mathematical identities, we get the wave equation for E field:
$ \nabla^2 \vec{E} + \omega^2\mu\epsilon(1-j\sigma/(\omega\epsilon))\vec{E} = 0 \space\space\space\space\space\space\space\space\space(Eq.3) $
If we check the definition of the <PERSON> equation por a generic vector wave $\vec{A}$,
$ \nabla^2 \vec{A} + k^2\vec{A} = 0 \space\space\space\space\space\space\space\space\space(Eq.4) $
we see that our expressions are much alike, and that we could define a propagation constant named $\gamma$ (corresponding to k in the expression of <PERSON> equation), which will have a real part $\alpha$, called the attenuation constant and an imaginary part $\beta$, the phase constant.
$\gamma=\alpha+j\beta\space = \omega\sqrt{\mu\epsilon}\sqrt{(1-j\sigma/(\omega\epsilon))} \space\space\space\space\space\space\space\space\space(Eq.5)$
In the definition of skin depth we were talking about a power decay, i.e, attenuation. 418
In fact, we can easily check that the skin depth (which from now on we will name $\delta_s$) corresponds to the inverse of the attenuation constant
$\delta_s=1/\alpha \space\space\space\space\space\space\space\space\space(Eq.6)$
Finally, given that $\alpha$ is the real part of the propagation constant $\gamma$, we can extract the expression of the skin depth by taking the real part of Eq. 5 and also assuming that, since the medium is a conductor, $\sigma>>\omega\epsilon$.
$ \delta_s = \sqrt{2/(\omega\mu\sigma)} $, which is, by the way, a distance measured in meters (although the very low values at high frequencies might persuade you to use millimeter of even microns)
Here you can see that this parameter is related to the conductivity of the media. By definition, this parameter gives you the distance at which the decay of the electric field is 1/e ~37%. This electric field is related to the surface current, as seen in Eq.2. 586