In this paper we introduce the class of two-point boundary-value descriptor systems (TPBVDS), discrete-time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system 'state' at the two endpoints of the system's interval of definition. By introducing a standard form for regular pencils we obtain a new and simple generalized Cayley-Hamilton theorem that simplifies our investigation of well-posedness, Green's function solutions, and reachability and observability for TPBVDS. There are two distinct notions of reachability and observability that one can define for TPBVDS, associated with processes that propagate inward from and outward toward the boundaries. We investigate each of these in detail, obtaining, among other things, far simpler forms for the reachability and observability results than found previously in the literature. In addition, we describe several methods for the efficient solution of TPBVDS, one involving recursions from each end of the interval toward the other and two others involving recursions that proceed outward toward and inward from the boundaries.