As far as I'm aware, a complex manifold $M$ is a topological manifold together with an atlas ${\cal A}$ of charts $(U_i,\varphi_i)\in{\cal A}$ such that the open sets $U_i$ cover $M$, the maps $\varphi_i: U_i\to \mathbb{C}^n$ are homeomorphisms and whenever $U_i\cap U_j\neq\emptyset$ the transition maps $\varphi_i\circ\varphi_j^{-1}:\mathbb{C}^n\to \mathbb{C}^n$ and $\varphi_j\circ\varphi_i^{-1}:\mathbb{C}^n\to \mathbb{C}^n$ are holomorphic. In that scenario, the atlas ${\cal A}$ is called a complex structure.
Now I'm studying string theory and in the book I'm using (Becker, Becker and Schwarz) what the authors call a complex structure seems like something different. Quoting their book, pages 89 and 90:
Here $\int Dh$ means the sum over all Riemann surfaces $(M,h)$. However, this is a gauge theory, since $S$ is invariant under diffeomorphisms and Weyl transformations. So one should really sum over Riemann surfaces modulo diffeomorphisms and Weyl transformations. Worldsheet diffeomorphism symmetry allows one to choose a conformally flat worldsheet metric $$h_{\alpha\beta}=e^\psi \delta_{\alpha\beta}.\tag{3.109}$$ When this is done, one must add the Faddeev-Popov ghost fields $b(z)$ and $c(z)$ to the worldsheet theory to represent the relevant Jacobian factors in the path integral. Then the local Weyl symmetry $(h_{\alpha\beta}\to \Lambda h_{\alpha\beta}$) allows one to fix $\psi$ (locally) - say to zero. However, this is not possible globally, due to a topological obstruction: $$\psi=0\Longrightarrow R(h)=0\Longrightarrow \chi(M)=0.\tag{3.110}$$ So, such a choice is only possible for worldsheets that admit a flat metric. Among orientable Riemann surfaces without boundary, the only such case is $n_{\rm h}=1$ (the torus). For ach genus $n_{\rm h}$ there are particular $\psi$s compatible with ${\chi}(M)=2-2n_{\rm h}$ that are allowed. A specific choice of such a $\psi$ corresponds to choosing a complex structure for $M$.
Now I'm puzzled with this last statement. This $\psi$ has to do with the metric, while the complex structure has to do with how we construct an atlas of holomorphic charts and therefore with how holomorphic functions are defined. I can't see how these two things are related.
What I imagined was this: we want to consider all possible Riemann surfaces $(M,h)$. Given any such surface the diffeomorphism symmetry allows the metric to be put in the form (3.109) and therefore be classified by $\psi$. Still, given a certain topology, not all metrics can be defined in the surface with that topology. Therefore we partition the space of such surfaces by genus $g$ and for each genus $g$ we have a class of possible $\psi$.
I'm conjecturing that the same happens with complex structure. In the same way that given a topology not all metrics can be defined in the surface with that topology, given a complex structure not all metrics are compatible with it.
Still I don't know if this is correct, and I think a more complete understanding is necessary here. So why does a specific choice of such a $\psi$ corresponds to choosing a complex structure for $M$?
As I said in the comment, what they wrote is simply nonsense. Here is a couple of references where at least you will not find any obvious nonsense:
Ji, Lizhen; Looijenga, Eduard J. N., Introduction to moduli spaces of Riemann surfaces and tropical curves, Surveys of Modern Mathematics 14. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-353-1/pbk). iv, 221 p. (2017). ZBL1375.14002.
The tropical staff is (probably) irrelevant for you and is discussed in Part II of the book.
Nag, Subhashis, Mathematics in and out of string theory, Kojima, Sadayoshi (ed.) et al., Topology and Teichmüller spaces. Proceedings of the 37th Taniguchi symposium, Katinkulta, Finland, July, 24–28, 1995. Singapore: World Scientific (ISBN 981-02-2686-1/hbk). 187-220 (1996). ZBL1050.32501.
A free version is available here, but there might be some differences with the published version.