We study the relations between stellar mass, star formation history, size and internal structure for a complete sample of 122,808 galaxies drawn from the Sloan Digital Sky Survey. We show that low-redshift galaxies divide into two distinct families at a stellar mass of 3 \times 10^10 M_sol. Lower mass galaxies have young stellar populations, low surface mass densities and the low concentrations typical of disks. A significant fraction of the lowest mass galaxies in our sample have experienced recent starbursts. At given stellar mass, the sizes of low mass galaxies are log- normally distributed with dispersion sigma(ln R_50) \sim 0.5, in excellent agreement with the idea that they form with little angular momentum loss through cooling and condensation in a gravitationally dominant dark matter halo. Their median stellar surface mass density scales with stellar mass as mu* propto M_*^0.54, suggesting that the stellar mass of a disk galaxy is proprtional to the three halves power of its halo mass. This suggests that the efficiency of the conversion of baryons into stars in low mass galaxies increases in propor- tion to halo mass, perhaps as a result of supernova feedback processes. At stellar masses above 3 \times 10^10 M_sol, there is a rapidly increasing frac- tion of galaxies with old stellar populations, high surface mass densities and high concentrations typical of bulges. In this regime, the size distribution is log-normal, but its dispersion decreases rapidly with increasing stellar mass and the median mass surface density is approximately constant. This suggests that the star formation efficiency decreases in the highest mass halos, and that little star formation occurs in massive galaxies once they have assembled.