The formation of nonrelativistic collisionless shocks in the laboratory with ultrahigh intensity lasers is studied via ab initio multidimensional particle-in-cell simulations. The microphysics behind shock formation and dissipation and the detailed shock structure are analyzed, illustrating that the Weibel instability plays a crucial role in the generation of strong subequipartition magnetic fields that isotropize the incoming flow and lead to the formation of a collisionless shock, similar to what occurs in astrophysical scenarios. The possibility of generating such collisionless shocks in the laboratory opens the way to the direct study of the physics associated with astrophysical shocks.