The inclusion of a nanoparticle into a polymer matrix is studied by efficient Monte Carlo simulations. The resulting structural changes in the melt and glass exhibit a strong dependence on the strength of the polymer attraction to the surface of the filler. The mechanical properties of the nanocomposite are analyzed in detail through a formalism that permits calculation of local elastic constants. The average shear and Young's modulus of the nanocomposite are higher than those of the pure polymer for neutral or attractive nanoparticles. For repulsive particles, these moduli are lower. Simulation of local properties reveals that a glassy layer is formed in the vicinity of the attractive filler, contributing to the increased strength of the composite material. In contrast, a region of negative moduli emerges around repulsive fillers, which provides a mechanism for frustration relief and a lowering of the glass transition temperature.