They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by Nagata (1962) who used them to show that algebraic varieties can be embedded in complete ones.
Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities.
This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme.
If S is the Zariski–Riemann space of a subring k of a field K, it has a topology defined by taking a basis of open sets to be the valuation rings containing a given finite subset of K. The space S is quasi-compact.
The valuation rings of a surface S over k with function field K can be classified by the dimension (the transcendence degree of the residue field) and the rank (the number of nonzero convex subgroups of the valuation group).