For simplicity, I'll just talk about varieties that are sitting in projective space or affine space. In algebraic geometry, you study varieties over a base field k. For our purposes, "over" just means that the variety is cut out by polynomials (affine) or homogeneous polynomials (projective) whose coefficients are in k.
Suppose that k is the complex numbers, C. Then affine spaces and projective spaces come with the complex topology, in addition to the Zariski topology that you'd normally give one. Then one can naturally give the points of a variety over C a topology inherited from the subspace topology. A little extra work (with the inverse function theorem and other analytic arguments) shows you that, if the variety is nonsingular, you have a nonsingular complex manifold. This shouldn't be too surprising. Morally, "algebraic varieties" are cut out of affine and projective spaces by polynomials, "manifolds" are cut out of other manifolds by smooth functions, and polynomials over C are smooth, and that's all that's going on.
In general, the converse is false: there are many complex manifolds that don't come from nonsingular algebraic varieties in this manner.
But in dimension 1, a miracle happens, and the converse is true: all compact dimension 1 complex manifolds are analytically isomorphic to the complex points of a nonsingular projective dimension-1 variety, endowed with the complex topology instead of the Zariski topology. "Riemann surfaces" are just another name for compact dimension 1 (dimension 2 over R) complex manifolds, and "curves" are just another name for projective dimension 1 varieties over any field, hence the theorem you described.
As for why Riemann surfaces are algebraic, Narasimhan's book explicitly constructs the polynomial that cuts out a Riemann surface, if you are curious.
You are trying to relate the periods of the curve (which are analytic invariants), to algebraic invariants, so the most you can get is some power series. Suppose you get from Gamma to the period matrix of the Jacobian: tau, then:
In genus 1 - which you are not interested in - you have the j-invariant, which is a function of tau.
In genus 2 you have the Igusa invariants.
In genus 3 you don't have a formula (too complicated), but you have an algorithm: there are 28 tangents to the theta divisor at the 28 odd 2-torsion points, these are the 28 bitangents of the canonical curve, and you can (effectively) reconstruct the curve from the bitangents.
In higher genera you can start in a similar way: map the 4-torsion points to some grassmanian (which is an embedding of A_g plus some level: Grushevsky and Salvati-Mani), but I'm not aware of a reconstruction algorithm.
Best Answer
The connection came from the paper by Dedekind and Weber "Theorie der algebraischen Functionen einer Veranderlichen", Crelle's Journal, 1882. In this paper the authors recover the theory by Riemann (including the famous Riemann-Roch theorem) by the abstract procedure of assigning a "curve" to an degree one transcendental field extension $\Omega/\mathbb{C}$ where "points" correspond to "places", i.e. discrete rank one valuations of the field $\Omega$. This theory inspired later further developments by Weil and Zariski and it may be considered the prehistory of the modern approach of abstract algebraic geometry.
This is also one of the first instances of modern structural mathematics based on (infinite) sets, much on the style later embraced by Bourbaki. The emphasis is on fields, valuations and ideals, the geometric counterpart of Dedekind's work in algebraic number theory. This point of view was not very well appreciated by some outstanding figures like Kronecker. This may be considered as a prelude of the controversy between formalists and intuitionists (Hilbert vs. Brouwer) that happened a few decades later.
There is an nice translation of Dedekind and Weber's paper by Stillwell published by the AMS, see AMS bookstore.