There are typically four(or more?) definition of meromorphic function on Riemann surface and three definitions of meromorphic function on complex manifolds, I want to show they are equivalent (I will not consider the additional one on Riemann surface as it's specified for that case):
Definition 1 Define the sheaf of meromorphic function to be $$\mathcal{M}_X=\coprod_{x \in X} \mathcal{M}_{X, x}$$ with topology that open sets are union of $\{G_x/H_x\mid x\in V\}\subset \mathcal{M}_X$ with $V$ open and $G,H \in \mathcal{O}_X(V)$. Meromorphic function is the section of the sheaf above.
Definition 2 the sheaf of meromorphic function is sheafification of the following sheaf:
Let $U$ be an open subset $U = \cup U_i$ for $U_i$ connected component then we have the presheaf $$U\mapsto \Pi_i \text{ Frac}(\mathcal{O}_X(U_i))$$
it's not hard to show the sheaf in definition 1is the Etale space assciated to this presheaf.
Definition 3 Let $U \subset \mathbb{C}^n$ be open. A meromorphic function $f$ on $U$ is a function on the complement of a nowhere dense subset $S \subset U$ with the following property: There exist an open cover $U=\bigcup U_i$ and holomorphic functions $g_i, h_i: U_i \rightarrow \mathbb{C}$ with $\left.\left.h_i\right|_{U_i \backslash S} \cdot f\right|_{U_i \backslash S}=\left.g_i\right|_{U_i \backslash S}$.
I was trying to show that definition 3 are equivalent to 1 (and 2), however I don't have good idea.
I see the point is the section of the sheaf, is a continuous map from $X\to \mathcal{M}_X$, with the topology, contains all the $$[(U,G/H)]=\{G_x/H_x \mid x \in U\}$$ being open in the etale space topology.
Therefore if $s:X\to \mathcal{M}_X$ such that $s_x = G_x / H_x$ then locally for where $G$ and $H$ are defined, $s_y = G_y/H_y$ for $y\in U$(that's guaranteed by the continuity of the section). Therefore, locally the information of the section is equivalent to the information of pair $(G,H)$, definition 3 use that locally information $(G_\alpha,H_\alpha)$ to define the section.
And you see that $U_i$ in definition 3 are the sets of open subsets that the representative element $(G,H)$ are defined.