the field of meromorphic functions
On a compact Riemann surface you cannot make interesting plain functions to C — the only ones are constants. So all the algebraic life of the surface lives in its MEROMORPHIC functions: those allowed to blow up at finitely many points. Collect every one of them and you get not just a set but a field — you can add, subtract, multiply, and divide them (away from poles) — and that single algebraic object turns out to remember the surface entirely.
Precisely, for a compact Riemann surface X let M(X) denote the set of all meromorphic functions on X. Sums, products, and (nonzero) quotients of meromorphic functions are meromorphic, the constants C sit inside as a subfield, and M(X) is a field containing C. The deep facts: M(X) always has transcendence degree exactly 1 over C (so it is a field of one variable's worth of freedom), it is finitely generated, and — the punchline — it is a finite extension of C(z), the rational functions in one variable. The Riemann sphere gives M = C(z) itself; a torus C/L gives C(wp, wp') with the cubic relation (wp')^2 = 4 wp^3 - g_2 wp - g_3 tying the two generators.
Why it matters: this is the bridge from analysis to algebra. The functor X -> M(X) is a perfect dictionary — two compact Riemann surfaces are isomorphic exactly when their function fields are isomorphic over C, and EVERY field of transcendence degree 1 finitely generated over C arises this way. That is why 'compact Riemann surface' and 'smooth projective algebraic curve over C' are two names for one thing. An honest subtlety: the equivalence is between the surface and the field as an ABSTRACT field over C; choosing a generator (a meromorphic function) is choosing a branched covering map to the sphere, and different generators give different presentations of the same field.
For the elliptic curve / torus with function field C(x, y) where y^2 = x^3 - x, the element x is a degree-2 map to the sphere (each value of x has two y's, the two square roots), so C(x, y) is a degree-2 extension of C(x). Recovering the surface from the field amounts to reading off this branched cover and where it ramifies.
y^2 = x^3 - x makes C(x,y) a degree-2 extension of C(x): the surface is a 2-sheeted cover of the sphere.
The dictionary is between compact Riemann surfaces and finitely generated transcendence-degree-1 fields over C. Non-compact surfaces (like the disk or the plane) have much larger, less rigid function fields and are NOT captured by this algebraic equivalence.