We report a computer-simulation study of the rate of homogeneous crystal nucleation and the structure of crystal nuclei in a Lennard-Jones system at moderate undercooling. The height of the nucleation barrier has been determined using umbrella sampling, whereas the barrier crossing rate is calculated using molecular dynamics simulation. The Simulations clearly show that the barrier crossing is a diffusive process. Nevertheless, the kinetic prefactor in the nucleation rate is found to be some two orders of magnitude larger than predicted by classical nucleation theory. The height of the barrier is in good agreement with the theoretical prediction. Although the Lennard-Jones system has a stable face-centered cubic (fee) phase below the melting line, the precritical nuclei are found to be mainly body-centered cubic (bcc) ordered. As they grow to their critical size, they become more fee ordered in the core. However, the critical and postcritical nuclei retain a high degree of bcc ordering in the interface. Furthermore it is found that in the interface the density falls off faster than the structural order parameter, which is in agreement with the predictions of density functional calculations. [P. Harrowell and D. W. Oxtoby, J. Chem. Phys. 89, 1639 (1984)].