Is it the same as the space of all possible descriptions of a single electron? If not, how do they differ?
Please give the mathematical name or specification of this space or these spaces.
Yes it is indeed. The Hilbert space of a single electron describes all possible states one electron can be in. It is (if we neglect spin) $L^2(R^3)$. (If we include spin, it becomes spinor valued wave functions instead of complex valued wave functions.)
There is a technicality: the zero vector of the Hilbert space does not describe a possible state, and, two vectors which differ by a scalar, describe the exact same physical state.
Amusingly, this Hilbert space is abstractly isomorphic to the Hilbert space of a deuteron, or a proton, or a meson, or.... actually, almost all Hilbert spaces of a finite system of distinguishable particles are abstractly isomorphic, but this has no practical importance, nor is it a theoretical nuisance, nor does it have much to do with your question.
I think Joseph f. johnson mixed up something. Not every two Hilbert spaces are isomorphic! For example, take $C^2$ and $C^3$, which are finite dimensional Hilbert spaces but not isomorphic.
What Joseph f. johnson might had in mind, was the following theorem: Let $H$ be a infinite dimensional Hilbert space (with some "nice" properties, eg separable), than one can always find a set $M$ and a measure $d \mu$ such that $H$ is isomorphic to $L^2(M, d \mu)$. So to say $L^2(M, d \mu)$ is the prototype for all Hilbert spaces. Examples: $L^2(R)$ is trivially isomorphic to $L^2(R)$, $L^2(R^3) \otimes L^2(R^3)$ is isomorphic to $L^2(R^6)$. The last one is the Hilbert space for 2 particles.
For the relativistic case one can create one-particle Hilbert spaces for particles with mass $m>0$. The base is than the space-like hyperboloid $k_\mu k^\mu = m, p_0 >0$ and the measure is $d^3 k / (2 k^0)$ in momentum representation.
The Hilbert space of an electron is the electron itself ! Even in classical mechanics we can replace the subjective notion of a system by your own phase space, this is not different in quantum mechanics.
The phase space in classical mechanics tells us how to build physical quantities: as a function defined in the phase space to the real line. In quantum mechanics the Hilbert Space tells us how to build observables: as linear hermitian operators.
So, for all purpose, the electron it is your own Hilbert Space. If you look in a symmetry way of looking, we have this idea more clear. The electron is a irreducible representation (this is a hilbert space) of some symmetries.
Let's go to the Hilbert Space (or to the electron ;) )
The Hilbert Space is defined in mathematical way here
The Hilbert Space in a physical perspective is a structure that embrace the superposition principle. You are able to understand the Hilbert Space as a physicist if you understand this principle. What this principle tells us is that the information contained in a quantum system are described by the length of a sum of a complex numbers. We need for Born principle for understanding this length of a complex numbers as a probability.
If we look for some Hilbert Space and fix some basis, we can identify for each vector basis a physical proposition. Define a state for your system as some vector, that is some linear combination of the vectors basis, we can interpret the probability of the proposition to be true as a projection of the state vector in the respective basis vector.