We describe the development of an FX-style correlator for very long baseline interferometry (VLBI), implemented in software and intended to run in multiprocessor computing environments, such as large clusters of commodity machines (Beowulf clusters) or computers specifically designed for high-performance computing, such as multiprocessor shared-memory machines. We outline the scientific and practical benefits for VLBI correlation, these chiefly being due to the inherent flexibility of software and the fact that the highly parallel and scalable nature of the correlation task is well suited to a multiprocessor computing environment. We suggest scientific applications where such an approach to VLBI correlation is most suited and will give the best returns. We report detailed results from the Distributed FX (DiFX) software correlator running on the Swinburne supercomputer (a Beowulf cluster of ~300 commodity processors), including measures of the performance of the system. For example, to correlate all Stokes products for a 10 antenna array with an aggregate bandwidth of 64 MHz per station, and using typical time and frequency resolution, currently requires an order of 100 desktop-class compute nodes. Due to the effect of Moore's law on commodity computing performance, the total number and cost of compute nodes required to meet a given correlation task continues to decrease rapidly with time. We show detailed comparisons between DiFX and two existing hardware-based correlators: the Australian Long Baseline Array S2 correlator and the NRAO Very Long Baseline Array correlator. In both cases, excellent agreement was found between the correlators. Finally, we describe plans for the future operation of DiFX on the Swinburne supercomputer for both astrophysical and geodetic science.