Jacobian of a Riemann surface and double unramified covers Take the jacobian of a riemann surface named $J(C)$ as the set of the line bundle of degree zero. Set $J_{2}(C)$ the subgroup of $J(C)$ of the element of orther two i.e. $L\in J(C)$ such that $L^{\bigotimes2}=O_C$.
The assertion is that there is a way to realize  $J_{2}(C)$ as the set $\Delta(C)$ of the double unramified cover under the Riemann surface $C$. How can I do it? 
 A: Let $A = Jac(C)$ be the Jacobian of a compact Riemann surface of genus $g$.
Let $A[2]$ be the points $x$ in $A$ such that $2x =0$ in $A$.
Let me try to sketch some of the ideas involved. I will leave the details for another day. (I hope I understood your question correctly btw.)
Note that, if $x$ is in $A[2]$, then there is a finite etale morphism $A\to A/<x>$. Let $A_x\to A$ be the dual of this isogeny. (Here we are using that the Jacobian is principally polarized to dualize isogenies.) Pull-back $A_x\to A$ to $C$ to get a finite etale morphism $D_x\to C$. 
The preceding paragraph induces a map from $A[2]$ to the set of double etale covers of $C$. Note that this map is well-defined. Moreover, note that $D_x\to C$ is trivial if and only if $x =0$. 
The inverse map can be constructed by running this construction backwards. So start with $D\to C$   a double etale cover of $C$. Assume $D$ is connected for simplicity. Then the universal property of $Jac(C)$ (as the Albanese of $C$ with respect to the choice of some base point on $C$) induces a morphism $Jac(D)\to Jac(C)$. The dual of this map gives an injective morphism of abelian varieties $Jac(C) \to Jac(D)$. Let $A_D$ be the image of this morphism. Then $Jac(C) \to A_D$ is an isogeny of degree two. Let $x_D$ be a generator of its kernel. The map $D\mapsto x_D$ is an inverse for the map $x\mapsto D_x$ constructed above.
BTW: you can show that the set of abelian (Galois) finite etale covers of $C$ of degree $d$ are in bijection with the set of finite etale covers of degree $d$ of $Jac(C)$. When $d=2$, it is a coincidence that this set is in bijection with $Jac(C)[2]$.
