What is the term for a three dimensional region of space - as we have surface and curve for two and one dimensions respectively?
I've sometimes seen this called a volume, although I find that misleading, as volume is typically a measure (a real number) on such a region.
Likewise for higher dimensions.
As someone answered you in comments, as far as I know, there is no specific terminology like in lower dimensions. However, one often refers to a manifold of dimension $3$ as 3-manifold or, in a fancier way, as threefold or $3$-fold.