We examine the origin and evolution of the mass-metallicity relationship (MZR, M sub(*)-Z) for galaxies using high-resolution cosmological smoothed particle hydrodynamics (SPH) + N-body simulations that include a physically motivated description of supernova feedback and subsequent metal enrichment. We discriminate between two sources that may contribute to the origin of the MZR: (1) metal and baryon loss due to gas outflow or (2) inefficient star formation at the lowest galaxy masses. Our simulated galaxies reproduce the observed MZR in shape and normalization at both z = 0 and 2. We find that baryon loss occurs due to UV heating before star formation turns on in galaxies with M sub(bar) < 10 super(8) M, but that some gas loss due to supernova-induced winds is required to subsequently reproduce the low effective chemical yield observed in low-mass galaxies. Despite this, we show that low star formation efficiencies, regulated by supernova feedback, are primarily responsible for the lower metallicities of low-mass galaxies and the overall M sub(*)-Z trend. We find that the shape of the MZR is relatively constant with redshift but that its normalization increases with time. Simulations with no energy feedback from supernovae overproduce metals at low galaxy masses by rapidly transforming a large fraction of their gas into stars. Despite the fact that our low-mass galaxies have lost a majority of their baryons, they are still the most gas-rich objects in our simulations due to their low star formation efficiencies.