We report the discovery of 6576 new spectroscopically confirmed white dwarf and subdwarf stars in the Sloan Digital Sky Survey Data Release 12. We obtain Teff, logg and mass for hydrogen atmosphere white dwarf stars (DAs) and helium atmosphere white dwarf stars (DBs), estimate the calcium/helium abundances for the white dwarf stars with metallic lines (DZs) and carbon/helium for carbon-dominated spectra (DQs). We found one central star of a planetary nebula, one ultracompact helium binary (AM CVn), one oxygen line-dominated white dwarf, 15 hot DO/PG1159s, 12 new cataclysmic variables, 36 magnetic white dwarf stars, 54 DQs, 115 helium-dominated white dwarfs, 148 white dwarf + main-sequence star binaries, 236 metal-polluted white dwarfs, 300 continuum spectra DCs, 230 hot subdwarfs, 2936 new hydrogen-dominated white dwarf stars, and 2675 cool hydrogen-dominated subdwarf stars. We calculate the mass distribution of all 5883 DAs with S/N>=15 in DR12, including the ones in DR7 and DR10, with an average S/N=26, corrected to the 3D convection scale, and also the distribution after correcting for the observed volume, using 1/Vmax.