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 � 10¹⁰ M_⊙. Lower-mass galaxies have young stellar populations, low surface mass densities and the low concentrations typical of discs. Their star formation histories are more strongly correlated with surface mass density than with stellar mass. 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 lognormally distributed with dispersion δ (ln R₅₀) ∼ 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 �_* ∝ M^0.54_*, suggesting that the stellar mass of a disc galaxy is proportional to the three halves power of its halo mass. All of this suggests that the efficiency of the conversion of baryons into stars in low-mass galaxies increases in proportion to halo mass, perhaps as a result of supernova feedback processes. At stellar masses above 3 � 10¹⁰ M_⊙, there is a rapidly increasing fraction of galaxies with old stellar populations, high surface mass densities and the high concentrations typical of bulges. In this regime, the size distribution remains lognormal, but its dispersion decreases rapidly with increasing mass and the median stellar mass surface density is approximately constant. This suggests that the star formation efficiency decreases in the highest-mass haloes, and that little star formation occurs in massive galaxies after they have assembled.