A normal toric variety corresponds to a strongly convex rational polyhedral fan in affine space. In this package, the fan associated to a normal $d$-dimensional toric variety lies in the rational vector space $\QQ^d$ with underlying lattice $N = \ZZ^d$. The fan is encoded by the minimal nonzero lattice points on its rays and the set of rays defining the maximal cones (a maximal cone is not properly contained in another cone in the fan). More information about the correspondence between normal toric varieties and strongly convex rational polyhedral fans appears in Subsection 3.1 of Cox-Little-Schenck.
Several methods for making new normal toric varieties from old ones are listed in the section on resolution of singularities.