We have carried out molecular dynamics simulations of the crystallization of hard spheres modeling colloidal systems that are studied in conventional and space-based experiments. We use microscopic probes to investigate the effects of gravitational forces, polydispersity, and of bounding walls on the phase structure. The simulations employed an extensive exclusive particle grid method and the type and degree of crystalline order was studied in two independent ways: by the structure factor, as in experiments, and through local rotational invariants. We present quantitative comparisons of the nucleation rates of monodisperse and polydisperse hard-sphere systems and benchmark them against experimental results. We show how the presence of bounding walls leads to wall-induced nucleation and rapid crystallization, and discuss the role of gravity on the dynamics of crystallization.