The r-process nucleosynthesis in core-collapse supernovae (CC-SNe) is studied, with a focus on the explosion scenario induced by rotation and strong magnetic fields. Nucleosynthesis calculations are conducted based on magneto-hydrodynamical explosion models with a wide range of parameters for initial rotation and magnetic fields. The explosion models are classified in two different types: prompt-magnetic-jet and delayed-magnetic-jet, for which the magnetic fields of proto-neutron stars (PNSs) during collapse and the core-bounce are strong and comparatively moderate, respectively. Following the hydrodynamical trajectories of each explosion model, we confirmed that r-processes successfully occur in the prompt-magnetic-jets, which produce heavy nuclei including actinides. On the other hand, the r-process in the delayed-magnetic-jet is suppressed, which synthesizes only nuclei up to the second peak (). Thus, the r-process in the delayed-magnetic-jets could explain only" weak r-process" patterns observed in metal-poor stars rather than the" main r-process," represented by the solar abundances. Our results imply that CC-SNe are possible astronomical sources of heavy r-process elements if their magnetic fields are strong enough, while weaker magnetic explosions may produce" weak r-process" patterns (). We show the potential importance and necessity of magneto-rotational SNe for explaining the galactic chemical evolution, as well as abundances of r-process enhanced metal-poor stars. We also examine the effects of the remaining uncertainties in the nature of PNSs due to weak interactions that determine the final neutron-richness of ejecta. Additionally, we briefly discuss radioactive isotope yields in primary jets (eg, 56 Ni), with relation to several optical observation of SNe and relevant high-energy astronomical phenomena.