See Algebraic Curves and Riemann Surfaces by Rick Miranda, page 87.
Definition
Let \(X\) and \(Y\) be Riemann surfaces and let \(p:X\to Y\) be a branched cover of degree \(d\). Consider a point \(q\) in a neighborhood \(U\) of a branch point chosen to be small enough so that it contains no other branch points. The fiber \(p^{-1}(q) = \{x_1,...,x_d\}\) of \(q\) then contains \(d\) points, and once we choose a basepoint \(x_i\), every loop \(\gamma \in \pi_1(U, q)\) lifts to a unique path \(\widetilde \gamma_i \) starting at \(x_i\). Likewise, every path from \(x_i\) to \(x_j\) maps to a unique class in \(\pi_1(U, q)\). We can therefore define a permutation \(\sigma_\gamma\) of \(\{x_1,...,x_d\}\) in the following way: \[\sigma_\gamma(x_i) = \widetilde \gamma_i(1).\] The map \[\rho:\pi_1(Y\setminus B, y_0) \to S_d, ~ \gamma \mapsto \sigma_\gamma\] is called the monodromy representation of \(p\).
Note: We can of course define this representation for any covering map \(f:X\to Y\); our definition uses an honest covering map, not a branched covering. However, in the case of Riemann surfaces, the monodromy representation is transitive.