The Galactic Archaeology with HERMES (GALAH) survey is a large-scale stellar spectroscopic survey of the Milky Way, designed to deliver complementary chemical information to a large number of stars covered by the Gaia mission. We present the GALAH second public data release (GALAH DR2) containing 342 682 stars. For these stars, the GALAH collaboration provides stellar parameters and abundances for up to 23 elements to the community. Here we present the target selection, observation, data reduction, and detailed explanation of how the spectra were analysed to estimate stellar parameters and element abundances. For the stellar analysis, we have used a multistep approach. We use the physics-driven spectrum synthesis of Spectroscopy Made Easy (SME) to derive stellar labels (Teff, logg, [Fe/H], [X/Fe], vmic, vsini, A_KS_) for a representative training set of stars. This information is then propagated to the whole sample with the data-driven method of The Cannon. Special care has been exercised in the spectral synthesis to only consider spectral lines that have reliable atomic input data and are little affected by blending lines. Departures from local thermodynamic equilibrium (LTE) are considered for several key elements, including Li, O, Na, Mg, Al, Si, and Fe, using 1D marcs stellar atmosphere models. Validation tests including repeat observations, Gaia benchmark stars, open and globular clusters, and K2 asteroseismic targets lend confidence to our methods and results. Combining the GALAH DR2 catalogue with the kinematic information from Gaia will enable a wide range of Galactic Archaeology studies, with unprecedented detail, dimensionality, and scope.