What is a "universal algebraic curve of genus 1"?

Question

I need to understand how exactly the construction of the universal algebraic curve (in this sense) goes in the context of elliptic curves over the complex numbers.

Answer 1

I'll answer your question for elliptic curves over $\mathbb{C}$. Note however that your link refers to $\mathcal{M}_1$, the moduli space of curves of genus $1$ (rather than elliptic curves, which need a marked point) - since automorphism groups of genus $1$ curves are infinite your question would become rather more subtle.

Anyway, for elliptic curves. First recall that elliptic curves $E$ and $F$ defined over an algebraically closed field are isomorphic if and only if they have the same $j$-invariant; from this it follows that $\mathcal{M}_{1,1}$ (or at least its $\mathbb{C}$-points, after passing to non-algebraically closed extensions one might need adjectives like "coarse space") is isomorphic to $\mathbb{A}^1_j$, we associate to each elliptic curve its $j$-invariant. Conversely for each $j \in \mathbb{C}$ we may associate the elliptic curve $$E_j : y^2 = x^3 - 27 j(j-1728)x - 54 j(j-1728)^2$$ which has $j$-invariant $j$.

The universal elliptic curve over $\mathcal{M}_{1,1}$ is then not so hard to define; it is the elliptic surface $$\mathcal{E} : y^2 = x^3 - 27j(j-1728)x - 54j(j-1728)^2$$ equipped with the obvious morphism $\mathcal{E} \to \mathcal{M}_{1,1}$.