Particle impact dampers (PIDs) have been shown to be effective in vibration damping. However, our understanding of such dampers is still limited, based on the theoretical models existing today. Although considerable research has been carried out in the generic field of 'impact oscillators,' much of it uses sophisticated mathematical tools and is somewhat inaccessible to the practicing engineer. Predicting the performance of the PID is an important problem, which needs to be investigated more thoroughly. This research seeks to understand the dynamics of a PID as well as what parameters govern its behavior. The system investigated is a particle impact damper with a ceiling, under the influence of gravity. The base is harmonically excited in the vertical direction. A discrete event approach is used, wherein the variables at one 'event' (or impact) uniquely dictate the variables at the next 'event', leading to a two-dimensional difference map. This map is then solved using a numerical continuation procedure. Periodic impact motions and 'irregular' motions are observed. The effects of various parameters such as the gap clearance, coefficient of restitution and the base acceleration are analyzed. The dependence of the loss factor on these parameters is also studied. The loss factor results indicate a peak for certain combinations of parameters. These combinations of parameters correspond to a region in parameter space where two-impact-per-cycle motions are observed over a wide range of non-dimensional base accelerations. The value of the non-dimensional acceleration at which the onset of two-impact-per-cycle solutions occurs depends on the non-dimensional gap clearance and the coefficient of restitution. The range of non-dimensional gap clearances over which two-impact-per-cycle solutions are observed increases as the coefficient of restitution increases. In the regime of two-impact-per-cycle solutions, the value of non-dimensional base acceleration corresponding to onset of these solutions initially decreases and then increases with increasing the non-dimensional gap clearance. As the two-impact-per-cycle solutions are associated with high loss factors that are relatively insensitive to changing conditions, they are of great interest to the designer.