A 2-sheeted surface is defined by the equation that you write,
$$
y^2=x(x-1)(x-a).
$$
Set $x=2^{2/3}X+\frac{a+1}{3}$. Then the equation becomes
$$
y^2=4X^3-h_2 X-h_3,
$$
for some constants $h_2$ and $h_3$ which are easy to determine.
A torus is typically parametrized by a coordinate $z$ with identifications
$$
z\sim z+1,\quad z\sim z+\tau, \qquad(1)
$$
where $\tau$ is the modulus of the torus. We would then like to find functions $\wp(z)$ and $\mathcal{Y}(z)$ such that
$$
\mathcal{Y}^2(z)=4\wp^3(z)-g_2 \wp(z)-g_3,
$$
and such that both $\mathcal{Y}$ and $\wp$ are periodic, i.e.
$$
\wp(z)=\wp(z+1)=\wp(z+\tau),\qquad(2)
$$
and same for $\mathcal{Y}(z)$. Provisionally $g_i=h_i$, but we will change this a bit below. If we manage to find such functions, we can then try to identify points on the 2-sheeted Riemann surface and on the torus as
$$
z\leftrightarrow (X,y)=(\wp(z),\mathcal{Y}(z)).
$$
Let us discuss the function $\wp(z)$. Remember that there is a branch point in our surface at $\infty$. We would thus like $\wp(z)$ to take value $\infty$ only once, but all the values in a neighborhood of $\infty$ twice (since there is a branch-cut). In other words, we want $\wp(z)$ to have only one singularity on the torus (modulo identifications (1)), but this singularity must be a double pole.
So we would like to find functions with periodicity property (2), and one double pole modulo (2). If we have such a function $\wp(z)$ then $A\wp(z+B)$ also works. We can fix this freedom by requiring that the double pole is at $z=0$ and has coefficient $1$, i.e.
$$
\wp(z)\sim z^{-2}, \quad(z\to 0).
$$
It turns out that these requirements completely fix $\wp$(z), and this function is called the Weierstrass's $\wp$-function. (This letter is "p".) See this Wikipedia page. It is implicit in the notation that I use, but $\wp(z)$ depends also on $\tau$.
One can then check that if we take $\mathcal{Y}(z)=\wp'(z)$, then
$$
\mathcal{Y}^2(z)=4\wp^3(z)-g_2 \wp(z)-g_3,\qquad (3)
$$
where $g_i=g_i(\tau)$ are determined by $\tau$. Since we only have one parameter $\tau$, we cannot generically solve two equations $g_i(\tau)=h_i$. However, we can rescale $X$ and $y$ as $y\to \lambda^{1/2}y$ and $X\to \lambda^{1/3} X$, which changes $h_2$ and $h_3$. It is only the ratio $h_2^{3}/h_3^2$ that is invariant under such rescalings and we only need to solve for
$$
h_2^3 h_3^{-2}=g_2^3(\tau)g_3^{-2}(\tau)
$$
to determine what $\tau$ is, and find a rescaling of $y$ and $X$ which will bring the equation to the form (3). This will determine the mapping between the 2-sheeted surface and the torus.
