This paper characterizes the radial structure of stellar population properties of galaxies in the nearby universe, based on 300 galaxies from the CALIFA survey. The sample covers a wide range of Hubble types, and galaxy stellar mass. We apply the spectral synthesis techniques to recover the stellar mass surface density, stellar extinction, light and mass-weighted ages, and mass-weighted metallicity, for each spatial resolution element in our target galaxies. To study mean trends with overall galaxy properties, the individual radial profiles are stacked in seven bins of galaxy morphology. We confirm that more massive galaxies are more compact, older, more metal rich, and less reddened by dust. Additionally, we find that these trends are preserved spatially with the radial distance to the nucleus. Deviations from these relations appear correlated with Hubble type: earlier types are more compact, older, and more metal rich for a given mass, which evidences that quenching is related to morphology, but not driven by mass. Negative gradients of ages are consistent with an inside-out growth of galaxies, with the largest ages gradients in Sb-Sbc galaxies. Further, the mean stellar ages of disks and bulges are correlated, with disks covering a wider range of ages, and late type spirals hosting younger disks. The gradients in stellar mass surface density depend mostly on stellar mass, in the sense that more massive galaxies are more centrally concentrated. There is a secondary correlation in the sense that at the same mass early type galaxies have steeper gradients. We find mildly negative metallicity gradients, shallower than predicted from models of galaxy evolution in isolation. The largest gradients occur in Sb galaxies. Overall we conclude that quenching processes act in manners that are independent of mass, while metallicity and galaxy structure are influenced by mass-dependent processes.