How do I approach this problem?
For a drawing of a graph $G$ on a torus one defines a 'toric dual' $G^{t*}$ of $G$ in the natural way: every face of $G$ corresponds to a vertex of $G^{t*}$ and every edge of $G$ corresponds to an edge of $G^{t*}$. Some graphs (like $K_5$ and $K_{3,3}$) have a toric dual but not dual. Explain why is toric dual in general not an abstract dual.
Hint: Remember that the dual of a graph $G$ only makes sense if $G$ is a plane graph (i.e. if you've fixed a particular embedding---if one exists!---of $G$ into the plane). Similarly, the toric dual of a graph $G$ only makes sense if $G$ is embedded in a torus.
If you've got a graph $G$ that can be embedded in a torus, does that necessarily mean it can be embedded in the plane? Try looking at your examples of $K_5$ and $K_{3,3}$.