I have a collection of points $P_1, ..., P_N$ in some Euclidean space $\mathbb R^m$ and the coordinates $x_1, x_2, ..., x_N$ respectively associated with them, where $x_i$ is the usual Cartesian tuple $x_i = (x_{i1}, x_{i2}, ..., x_{im})$. I can immediately calculate the pairwise distances between these points $r_{12}, r_{13}, ..., r_{N-1,N}$ under the usual Euclidean norm using Pythagoras' theorem (in $m$ dimensions), i.e. $r_{ij} = \left\Vert x_i - x_j \right\Vert$.

Suppose now I have the converse situation where I have the points $P_1, ..., P_N$ and all their associated pairwise distances $\{r_{ij}\}$, and I don't know their coordinates. What is known about the embeddability of these points in an Euclidean space, and is it possible to reconstruct the Cartesian coordinates for them? In other words, given an arbitrary collection of nonnegative numbers $\{r_{ij}\}$, how do I find all positive integers $m$ and enumerate all the possible sets of $N$ coordinates $x_1, x_2, ..., x_N \in \mathbb R^m$ that are consistent with the interpretation $r_{ij} = \left\Vert x_i - x_j \right\Vert$?