We investigate the effect of an electric field on a highly charged polyelectrolyte, (PE) chain by Brownian-dynamics simulation methods and a combination of different scaling arguments. In the simulation we take the counterions explicitly into account and therefore include the effects of counterion condensation and PE collapse as the coupling parameter (proportional to counterion valency) is increased. For highly charged PEs in the collapsed phase, a norrequilibrium unfolding transition occurs at sufficiently high electric fields where the PE aligns parallel to the external field. The critical field strength E* is determined from scaling results for the polarizability of a PE globule and exhibits a dependence on the chain length N, E* similar to N-1/2, which might be useful for electrophoretic separation of charged, collapsed biopolymers. For noncollapsed PEs this unfolding transition is less pronounced: the critical field depends on the swelling exponent v and scales as E* similar to N-3v/2 The electrophoretic mobility of PE monomers and counterions is determined. For large fields, counterions bound to the PE contribute significantly to the total conduction, since they can glide along the PE. This is an important factor in understanding experimental conduction experiments. The electrophoretic mobility of the bound counterions is determined by the electrostatic friction with the PE backbone and an activation barrier for decondensing from the PE; it in fact changes sign as the field strength is increased.