In this paper, we study the entire solutions of a discrete diffusive equation with inhomogeneous bistable nonlinearity. Here, the inhomogeneous nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. An entire solution means a classical solution defined for all space and time variables. It is known that the homogeneous bistable equation has unique bistable traveling wavefront. We first give the asymptotic estimates of the bistable traveling wavefront of the homogeneous bistable equation and then establish the comparison principle for the inhomogeneous bistable equation. Next, based on the bistable traveling wavefront of homogeneous equation, we construct some suitable super- and sub-solutions. By applying the comparison principle and the super- and sub-solutions method, we prove the existence of entire solutions. Moreover, the strict monotonicity in time and uniqueness of entire solutions are obtained. [ABSTRACT FROM AUTHOR]