The magnetoconductivity s(B) in the two-dimensional (2D) nondegenerate electron fluid and 2D solid has been analyzed theoretically and investigated experimentally, from 60 mK to 1.3 K in magnetic fieldsB up to 8 Tesla. In the fluid phase, s(B) is described by the Drude model in weak to moderately strong classical fields, including the range ßB»1. At higher fields (depending on the density s(B) is nonmonotonous and diplays a minimum. This behavior is due to many-electron effects, which can be described in terms of cyclotron orbit diffusion controlled by an internal fluctuational electric field. The squared internal field derived from experiments is in good agreement with computer simulations. In the solid phase electron transport becomes strongly non-linear even for weak driving voltagesV 0. Experimentally we determine, from the losses, the effective AC Corbino conductivity at a frequencyf. We find that s(BafV 0/B forV 0 below some threshold voltageV c . In this region the Hall velocity ¿ H approaches the ripplon phase velocityv 1=w(G 1)/G 1 at the first reciprocal lattice vectorG 1 of the electron solid. We suggest that this behaviour is due to to a resonant drag force from the Bragg-Cerenkov radiation of coherent ripplons by the moving crystal.