A good starting place would be to read about branched coverings of Riemann surfaces, which are explained in many textbooks on the subject.
A surjective map of Riemann surfaces $f : X \to Y$ is a branched covering if for each $y \in Y$ there exists a complex coordinate chart $U$ around $y$ parameterized by $z < \epsilon$, where $0 < \epsilon < 1$, such that for each component $V$ of $f^{-1}(U)$ there exists an integer $k \ge 1$ and a complex coordinate chart for $V$ parameterized by $w < \delta$, where $\delta = \epsilon^{1/k}$, such that the map $f : U \to V$ is given in coordinates by $z=w^k$.