Discrete Differential Geometry aims to develop a discrete theory which respects fundamental aspects of smooth theory. With this in mind, some results of smooth theory of Affine Geometry are firstly introduced since their discrete counterparts shall be treated a posteriori. The first goal of this work is construct a discrete affine structure for nets in a three-dimensional space with indefinite Blaschke metric and asymptotic parameters. For this purpose, one defines a conormal vector field, which satisfies Lelieuvre s equations and it is associated to a real parameter; and an affine normal vector field, which defines the cubic form of the net and makes the structure well defined. This structure allows to study, e.g., ruled surfaces with emphasis on improper affine spheres. Moreover, a definition for singularities is proposed in the case of discrete improper affine spheres from the center-chord construction. Another goal here is to propose a definition for an asymptotic net with constant affine mean curvature (CAMC), in a way that encompasses discrete affine minimal surfaces and discrete affine spheres. Discrete affine minimal surfaces receive a beautiful geometrical characterization directly linked to discrete Lie quadrics. This work is completed with the main result about a discrete version of Cayley surfaces, which are ruled improper affine spheres that can be characterized by the induced connection as: an asymptotic net with CAMC is equiaffinely congruent to a Cayley surface if and only if the cubic form does not vanish and the affine induced connection is parallel.