What is the Hilbert space of a single electron? 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.
 A: 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.
A: 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.
