Let $X$ be a compact Riemann surface, and $f: X \to \mathbb{P}^1(\mathbb{C})$ be a holomorphic branched cover. This induces a finite extension of their fields of meromorphic functions $f^*: \mathbb{C}(z) \to \mathcal{M}(X)$. $\mathbb{C}(z)$ is the field of fractions of $\mathbb{C}[z]$, which is a principal ideal domain, and therefore a Dedekind domain. The integral closure $\mathcal{O}$ of $\mathbb{C}[z]$ in $\mathcal{M}(X)$ is also a Dedekind domain. I believe one can show that $\mathcal{O} = \{g \in \mathcal{M}(X): g^{-1}(\infty) \subseteq f^{-1}(\infty)\}$, so it consists of the functions which are finite everywhere the cover $f$ is finite.
What can we say about the class group of $\mathcal{O}$? Is it related to the Picard group of $X$? As I understand, there is a distinction between the two, in that the prime ideals of $\mathcal{O}$ should correspond to points in $f^{-1}(\mathbb{P}^1 - \infty)$. So the ideal structure of $\mathcal{O}$ does not "see" the fiber of $f$ over $\infty$, whereas the Picard group accounts for arbitrary divisors on $X$. Different choices of $f$ will gives us different integral closures of $\mathbb{C}[z]$ in $\mathcal{M}(X)$. Are there any properties which are invariant under choice of cover?
See Forster, Lectures on Riemann Surfaces, chapter 8 for the algebraic properties of branched covers of Riemann surfaces.