A fan $\Delta$ is a collection of cones closed under the operations of taking faces and intersections. Each cone gives rise to an affine variety. The result of gluing these along intersections gives the toric variety of this fan $X_\Delta$.

This correspondence extends functorially. Fan morphisms between a fan $\Delta_1$ in $N_1$ to $\Delta_2$ in $N_2$ is a linear map $f$ from $N_1$ to $N_2$ such that every cone $\sigma \in \Delta_1$ goes to $f (\sigma) \subset \sigma_2$ where $\sigma_2$ is a cone in $\Delta_2$.

Orbit-Cone Correspondence

References

Ezra Miller, What is… a toric variety?, Notices of the AMS, volume 55, number 5 (pdf)

Pavel Dimitrov, Toric varieties, a short introduction (pdf)