We present spectroscopic metallicities of individual stars in seven gas-rich dwarf irregular galaxies (dIrrs), and we show that dIrrs obey the same mass-metallicity relation as the dwarf spheroidal (dSph) satellites of both the Milky Way and M31: Z_* ~ M_*^(0.30 +/- 0.02). The uniformity of the relation is in contradiction to previous estimates of metallicity based on photometry. This relationship is roughly continuous with the stellar mass-stellar metallicity relation for galaxies as massive as M_* = 10^12 M_sun. Although the average metallicities of dwarf galaxies depend only on stellar mass, the shapes of their metallicity distributions depend on galaxy type. The metallicity distributions of dIrrs resemble simple, leaky box chemical evolution models, whereas dSphs require an additional parameter, such as gas accretion, to explain the shapes of their metallicity distributions. Furthermore, the metallicity distributions of the more luminous dSphs have sharp, metal-rich cut-offs that are consistent with the sudden truncation of star formation due to ram pressure stripping. ; Comment: 24 pages, 13 figures, 1 machine-readable table included in "Other formats," accepted to ApJ. Minor typographical corrections in v.2