Various different physical processes contribute to the star formation and stellar mass assembly histories of galaxies. One important approach to understanding the significance of these different processes on galaxy evolution is the study of the stellar population content of today's galaxies in a spatially resolved manner. The aim of this paper is to characterize in detail the radial structure of stellar population properties of galaxies in the nearby universe, based on a uniquely large galaxy sample, considering the quality and coverage of the data. The sample under study was drawn from the CALIFA survey and contains 300 galaxies observed with integral field spectroscopy. These cover a wide range of Hubble types, from spheroids to spiral galaxies, while stellar masses range from M⋆ ~ 109 to 7 × 1011 M⊙. We apply the fossil record method based on spectral synthesis techniques to recover the following physical properties for each spatial resolution element in our target galaxies: the stellar mass surface density (μ⋆), stellar extinction (AV), light-weighted and mass-weighted ages (⟨log age⟩L, ⟨log age⟩M), and mass-weighted metallicity (⟨log Z⋆⟩M). To study mean trends with overall galaxy properties, the individual radial profiles are stacked in seven bins of galaxy morphology (E, S0, Sa, Sb, Sbc, Sc, and Sd). We confirm that more massive galaxies are more compact, older, moremetal 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 M⋆, which is evidence that quenching is related to morphology, but not driven by mass. Negative gradients of ⟨log age⟩L are consistent with an inside-out growth of galaxies, with the largest ⟨log age⟩L gradients in Sb-Sbc galaxies. Further, the mean stellar ages of disks and bulges are correlated and with disks covering a wider range of ages, and late-type spirals hosting younger disks. However, age gradients are only mildly negative or flat beyond R ~ 2 HLR (half light radius), indicating that star formation is more uniformly distributed or that stellar migration is important at these distances. The gradients in stellar mass surface density depend mostly on stellar mass, in the sense that more massive galaxies are more centrally concentrated. Whatever sets the concentration indices of galaxies obviously depends less on quenching/morphology than on the depth of the potential well. There is a secondary correlation in the sense that at the same M⋆ early-type galaxies have steeper gradients. The μ⋆ gradients outside 1 HLR show no dependence on Hubble type. We find mildly negative ⟨log Z⋆⟩M gradients, which are shallower than predicted from models of galaxy evolution in isolation. In general, metallicity gradients depend on stellar mass, and less on morphology, hinting that metallicity is affected by both - the depth of the potential well and morphology/quenching. Thus, the largest ⟨log Z⋆⟩M gradients occur in Milky Way-like Sb-Sbc galaxies, and are similar to those measured above the Galactic disk. Sc spirals show flatter ⟨log Z⋆⟩M gradients, possibly indicating a larger contribution from secular evolution in disks. The galaxies from the sample have decreasing-outward stellar extinction; all spirals show similar radial profiles, independent from the stellar mass, but redder than E and S0. 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. Appendices are available in electronic form at http://www.aanda.org