Let $X$ be the compact Riemann surface and $H$ a finite subgroup of $G=Aut(X)$. Then you can think of two different constructions.

 1. One is taking the quotient in the category of complex manifolds $X/H$ (which corresponds to the [GIT quotient][1] $X//H$ in the algebraic category). Quotients (of complex manifolds by complex Lie groups) don't always exist, so you have to *prove* that there actually exists a complex manifold $X/H$ which has the desired quotient properties (which is done in your textbook).  
 2. The other construction is considering the quotient *orbifold* $[X/H]$. In this case, since $H$ is finite, it has always a meaning, and is -indeed- a 1-dimensional orbifold in the complex manifolds category. 

The relation between the two constructions is that (in this case of smooth Riemann surfaces and finite groups) you can think of $[X/H]$ as the Riemann surface $X/H$ decorated, at the ramification points of the quotient map $\pi:X\to X/H$, with the stabilizers. 

  [1]: http://en.wikipedia.org/wiki/Geometric_invariant_theory