In Hörmander's text an "Introduction to Several Complex Variables," he gives the following definition for a Riemann domain on page 139:
A complex manifold $\Omega$ of dimension $n$ is called a Riemann domain if analytic functions separate points on $\Omega$ and there is an analytic map \begin{equation*} \varphi : \Omega \to \mathbb{C}^n \end{equation*} which is everywhere regular, that is, locally an isomorphism.
Edit: many authors drop the word analytic and only say $\varphi$ need be a local homeomorphism.
I understand the condition concerning separating points but what I don't understand is why this second condition ($\varphi$ locally an isomorphism) is needed on top of the usual complex manifold definition. Why can we not just patch together the homeomorphisms given from the definition of a complex manifold using a partition of unity and take that as our $\varphi$? In other words, what are we trying to convey that doesn't come from the usual definition of a complex manifold?
In a similar vein, for a Riemann surface, why is such a condition not necessary? We merely define them as one-dimensional complex manifolds without stating that there need exist such a $\varphi$.