How do we formally define a Riemann Surface in the context of complex analysis?
I know the basic principal of what it is (the joining together along branch cuts of different branches of a function so at each point on a surface $f(z)$ is uniquely defined). But I am looking for a more formal definition and the conditions when a Riemann surface can and can't be defined?
A Riemann surface is a one-dimensional complex manifold. Here is the formal definition :
Let $X$ be a Hausdorff topological space. A local complex chart of $X$ is a couple $(U,\phi)$, where $U$ is an open set of $X$ and $\phi$ an homeomorphism between $U$ and an open set $\Omega$ of $\mathbb{C}^n$ for a certain $n \in \mathbb{N}$.
Two local complex charts are $(U_1,\phi_1)$ and $(U_2,\phi_2)$ are said to be compatible if $$\phi_2 \circ \phi_1^{-1} : \phi_1(U_1 \cap U_2) \to \phi_2(U_1 \cap U_2)$$ is holomorphic.
A holomorphic atlas on $X$ is a set of local complex charts such that any pair is compatible and the domain of the charts cover $X$. An atlas is saturated if it is maximal for the inclusion.
A complex manifold is a Hausdorff topological space $X$ equiped with a holomorphic saturated atlas.
Now consider a complex manifold $X$ and some point $x \in X$. By definition, there is a chart $(U,\phi)$ with $x \in U$ and such that $\phi(U)$ is an open set of $\mathbb{C}^n$. You can check that the number $n$ is independant of the chosen chart. Hence $n$ depends only of $x$. We call it the complex dimension at the point $x$.
Finally, a Riemann surface is a complex manifold of dimension $1$ at each point.
An example is $\mathbb{C}$ itself or the Alexandroff compactification $\overline{\mathbb{C}}$ with its two charts $$\phi_1 : \mathbb{C} \to \mathbb{C} : z \mapsto z,$$ and $$\phi_2 : \overline{\mathbb{C}}_0 \to \mathbb{C} : z \mapsto \frac{1}{z}.$$
Now why do we use Riemann surfaces to talk about multivalued functions like the complex logarithm ? That's because a multivalued function on $\mathbb{C}$ gives a traditionnal function between Riemann surfaces. Here is a picture of the Riemann surface associated to the logarithm https://en.wikipedia.org/wiki/Complex_logarithm#/media/File:Riemann_surface_log.svg.