The word moduli (Moduln in German) first appeared in 1857 in Bernhard Riemann's celebrated paper "Theorie der Abel'schen Functionen".
The term moduli is also used in string theory to refer to various continuous parameters that label possible string backgrounds: the expectation value of the dilaton field, the parameters (e.g. the radius and complex structure) which govern the shape of the compactification manifold, et cetera.
A notable exception occurs when the various vacua in question are related by a symmetry which guarantees that their energy levels remain exactly degenerate.
N=1 means that the fermionic components of the supersymmetry algebra can be assembled into a single Majorana supercharge.
The total moduli space is locally a product of these two branches, as nonrenormalization theorems imply that the metric of each is independent of the fields of the other multiplet.
In the case of global N=2 supersymmetry, in other words in the absence of gravity, the Coulomb branch of the moduli space is a special Kähler manifold.
The Higgs branch is a hyperkähler manifold as was shown by Luis Alvarez-Gaume and Daniel Freedman in their 1981 paper Geometrical Structure and Ultraviolet Finiteness in the Supersymmetric Sigma Model.
Jonathan Bagger and Edward Witten demonstrated in their 1982 paper Matter Couplings in N=2 Supergravity that in this case, the Higgs branch must be a quaternionic Kähler manifold.