Discrepancies have emerged between the predictions of standard cold dark matter (CDM) theory and observations of clustering on subgalactic scales. Warm dark matter (WDM) is a simple modification of CDM in which the dark matter particles have initial velocities due either to their having decoupled as thermal relics or to their having been formed via nonequilibrium decay. We investigate the nonlinear gravitational clustering of WDM with a high-resolution N-body code and identify a number of distinctive observational signatures. Relative to CDM, halo concentrations and core densities are lowered, core radii are increased, and large halos emerge with far fewer low-mass satellites. The number of small halos is suppressed, and those present are formed by "top-down" fragmentation of caustics, as part of a "cosmic web" connecting massive halos. Few small halos form outside this web. If we identify small halos with dwarf galaxies, then their number, spatial distribution, and formation epoch appear in better agreement with the observations for WDM than they are for CDM.