Why is a manifold called a manifold?

Manifold is simply a fancy word for variety or plurality.

Before Riemannian (or non-Euclidean) geometry, only the flat kind of spaces was considered, aka Euclidean spaces, for each dimension. So, for instance, when a $2$-D space was concerned, it was only a flat plane and that was the only kind of space (same goes to $3$-D, etc.) Then Mathematicians (like Riemann) started to think of curved-kinds, which made an $n$-dimensional space no longer of a unique, flat kind but variety or Mannigfaltigkeit (ie, manifold in English) of dimension $n$.