The low-temperature antiferroelectric (AFE) phase of NH4H2PO4 corresponds to H ordering in O-H-O bridges leading to H2PO4 group polarizations perpendicular to the tetragonal c axis and alternating in chains. We determine the microscopic origin of such order by means of first-principles calculations in the framework of the density functional theory. The formation of N-Hcdots, three dots, centeredO bridges with correlated charge transfers and NH4+ group distortions turn out to be essential in stabilizing the AFE configuration against a c-polarized ferroelectric (FE) phase, as well as other FE states polarized perpendicular to the c axis. These FE states lie only a few meV above the AFE phase, which explains the observation of FE-AFE phase coexistence near the AFE transition.