The original source for the modern version is
H.Grauert and R. Remmert, Komplexe RaumeRäume, Math.Ann. 136 (1958), 245–318.
I don't know of an exposition in English.
The version due to Riemann was just for algebraic curves. This is covered in many sources; my favorite is Narasimhan's book "Compact Riemann Surfaces".