The production of J/psi mesons with rapidity 1.5 < y < 4.0 or 5.0 < y < 2.5 and transverse momentum PT < 14 GeV/e is studied with the LHCb detector in proton-lead collisions at a nucleon-nucleon centre-of-mass energy, root(NN)-N-S = 5 TeV. The J/psi mesons are reconstructed using the dimuon decay mode. The analysis is based on a data sample corresponding to an integrated luminosity of about 1.6 nb-1. For the first time the nuclear modification factor and forward-backward production ratio are determined separately for prompt J/psi mesons and J/psi from b-hadron decays. Clear suppression of prompt J/psi production with respect to proton-proton collisions at large rapidity is observed, while the production of J/psi from b-hadron decays is less suppressed. These results show good agreement with available theoretical predictions. The measurement shows that cold nuclear matter effects are important for interpretations of the related quark-gluon plasma signatures in heavy-ion collisions.