A (elementary) summary of the relation between algebraic curves and Riemann Surfaces

Question

I am interested in researching the links between these two topics but have no experience with Riemann Surfaces. What does it mean for a surface to be Riemann type and why does this link to algebraic curves.

Answer 1

I think that the two links in the comment of Moishe Kohan will give you more information, but let me tell a little about it.

Riemann surfaces are by definition complex manifold of $\mathbb{C}$-dimension $1$. Thus they are of $\mathbb{R}$-dimension $2$, that's why they are (initially) called "surfaces".

Now, a theorem states that any compact Riemann surface can be embedded into $\mathbb{P}^n_\mathbb{C}$ for some $n\geq 1$ (in fact, $n=3$ works for all of them!). Then Chow's theorem tells you that any closed submanifold $X\subset\mathbb{P}^n$ is algebraic, that is $X$ is given as the zero locus of homogeneous polynomials. So this way you can see that all compact Riemann surfaces are algebraic.

Conversely, if you consider an algebraic curve in $C\subset\mathbb{P}^n_\mathbb{C}$, given as the zero locus of some homogeneous polynomials, and if $C$ has no singularities (which is translated by a criterion on the rank of the jacobian matrix of the polynomials) then $C$ inherits a structure of Riemann surface : on the charts of $\mathbb{P}^n_\mathbb{C}$, the curve $C$ is given as the zero locus of polynomials, which are holomorphic.

If you want to go further in this "$\mathbb{C}$-manifold / $\mathbb{C}$-variety" correspondance, you should give a look at the really beautiful Serre's GAGA principle.

Another direction of interest, linked with your question, might be "When does a real surface (or manifold) admits a complex structure ?" which is an important question, for example in symplectic geometry (see e.g. https://en.wikipedia.org/wiki/Almost_complex_manifold).