Other posters explain some of the topology of spin structures. Here's a differential-geometric answer relevant to Dirac operators. The exercise you have set yourself, of understanding Dirac operators on the 2-torus, is a good one. Rather than trying to do it for you, I'll instead discuss the 3-torus (cf. Kronheimer-Mrowka, "Monopoles and 3-manifolds").

So: a spin-structure on a Riemannian 3-manifold $Y$ can be understood in the following workmanlike way: we give a rank 2 hermitian vector bundle $S \to Y$ (the spinor bundle); a unitary trivialization of $\Lambda^2 S$; and a Clifford multiplication map $\rho \colon TY\to \mathfrak{su}(S)$, such that at each $y\in Y$ there is some oriented orthonormal basis $(e_1,e_2,e_3)$ for $T_y Y$ such that $\rho(e_i)$ is the $i$th Pauli matrix $\sigma_i$. More invariantly, one can instead say that $\rho$ is an isometry (with respect to the inner product $(a,b)=tr(a^\ast b)/2$) and satisfies the orientation condition $\rho(e_1)\rho(e_2)\rho(e_3)=1$.

If we have two spin-structures, with spinor bundles $S$ and $S'$, we can look at the sub-bundle of $\mathrm{SU}(S,S')$ consisting of those fibrewise special isometries that intertwine the Clifford multiplication maps. This bundle has fibre $\{ \pm 1 \}$: it is a 2-fold covering of $Y$. As such it is classified by a class in $H^1(Y;\mathbb{Z}/2)$, whose non-vanishing is clearly the only obstruction to isomorphism of the two spin-structures. Conversely, by tensoring everything by real orthogonal line bundles (work out what this means concretely!), you can construct all spin structures, up to isomorphism, from a chosen one.

On flat $T^3$, all the data can be taken translation invariant. The Dirac operator is then $D = \sum_i{\sigma_i\partial_i}$. Tensoring with an orthogonal line bundle $\lambda$ (constructed, if you will, from a character $\pi_1(T^3)\to O(1)$) the formula becomes $D_\lambda =D \otimes 1_\lambda$.

In 2 dimensions, the story will be similar; the new feature is that the spinor bundle splits into two line bundles. The translation-invariant Dirac operator is nothing but the Cauchy-Riemann operator $\partial/\partial x + i \partial/\partial y$.