This is, of course, a long story incorporating many strands but I will try to give a quick overview.  Firstly, it is, as so often, convenient to skip to a more general framework.  In your  case, this would be that of an unbounded self-adjoint operator $T$ on Hilbert space (here that would be the Laplacian--more about that later).

One can associate with it a Frechet space  $H^\infty(T)$ (the intersection of the domains of definitions of its powers) and a  $DF$-space $H^{-\infty}(T)$ which are in duality.  In the case of classical differential operators (the most frequent examples occur with the Laplacian and Schrödinger operators), the former is a space of test functions, the latter of distributions.

If the spectrum of $T$ is discrete and consists of a sequence $(\lambda_n)$ of eigenvalues which are such that $|\lambda_n|$ is asymptotically like some positive $n^\alpha$ for some positive $\alpha$, then the situation is particularly transparent.  $H^\infty$ and $H\{-\infty}$ are a nuclear Frechet space and Silva space respectively.  The eigenfunctions of $T$ form a basis for both spaces and they are identifiable with  the sequence spaces $s$ and $s´$ of rapidly decreasing resp. slowly increasing sequences.

The smoothing operators, i.e., continuous linear operators from $H^{-\infty}$ into 
$H^\infty$ are then identified with the the two variable version of $s$ in a standard way.