  We study the radial structure of the stellar mass surface density ($\mu$) and
stellar population age as a function of the total stellar mass and morphology
for a sample of 107 galaxies from the CALIFA survey. We use the fossil record
to recover the star formation history (SFH) in spheroidal and disk dominated
galaxies with masses from 10$^9$ to 10$^{12}$ M$_\odot$. We derive the half
mass radius, and we find that galaxies are on average 15% more compact in mass
than in light. HMR/HLR decreases with increasing mass for disk galaxies, but is
almost constant in spheroidal galaxies. We find that the galaxy-averaged
stellar population age, stellar extinction, and $\mu$ are well represented by
their values at 1 HLR. Negative radial gradients of the stellar population ages
support an inside-out formation. The larger inner age gradients occur in the
most massive disk galaxies that have the most prominent bulges; shallower age
gradients are obtained in spheroids of similar mass. Disk and spheroidal
galaxies show negative $\mu$ gradients that steepen with stellar mass. In
spheroidal galaxies $\mu$ saturates at a critical value that is independent of
the galaxy mass. Thus, all the massive spheroidal galaxies have similar local
$\mu$ at the same radius (in HLR units). The SFH of the regions beyond 1 HLR
are well correlated with their local $\mu$, and follow the same relation as the
galaxy-averaged age and $\mu$; suggesting that local stellar mass surface
density preserves the SFH of disks. The SFH of bulges are, however, more
fundamentally related to the total stellar mass, since the radial structure of
the stellar age changes with galaxy mass even though all the spheroid dominated
galaxies have similar radial structure in $\mu$. Thus, galaxy mass is a more
fundamental property in spheroidal systems while the local stellar mass surface
density is more important in disks.
