We present the second major release of data from the Sydney - Australian Astronomical Observatory Multi-Object Integral Field Spectrograph (SAMI) Galaxy Survey. Data Release Two includes data for 1559 galaxies, about 50 per cent of the full survey. Galaxies included have a redshift range 0.004 < z < 0.113 and a large stellar mass range 7.5 < log (M*/M⊙) < 11.6. The core data for each galaxy consist of two primary spectral cubes covering the blue and red optical wavelength ranges. For each primary cube, we also provide three spatially binned spectral cubes and a set of standardized aperture spectra. For each core data product, we provide a set of value-added data products. This includes all emission line value-added products from Data Release One, expanded to the larger sample. In addition, we include stellar kinematic and stellar population value-added products derived from absorption line measurements. The data are provided online through Australian Astronomical Optics' Data Central. We illustrate the potential of this release by presenting the distribution of ~350 000 stellar velocity dispersion measurements from individual spaxels as a function of R/Re, divided in four galaxy mass bins. In the highest stellar mass bin [log (M*/M⊙) > 11], the velocity dispersion strongly increases towards the centre, whereas below log (M*/M⊙) < 10 we find no evidence for a clear increase in the central velocity dispersion. This suggests a transition mass around log (M*/M⊙) ~ 10 for galaxies with or without a dispersion-dominated bulge.