We present ultraviolet, optical, and near-infrared photometry as well as optical spectra of the peculiar supernova (SN) 2008ha. SN 2008ha had a very low peak luminosity, reaching only M_V = -14.2 mag, and low line velocities of only ~2000 km/s near maximum brightness, indicating a very small kinetic energy per unit mass of ejecta. Spectroscopically, SN 2008ha is a member of the SN 2002cx-like class of SNe, a peculiar subclass of SNe Ia; however, SN 2008ha is the most extreme member, being significantly fainter and having lower line velocities than the typical member, which is already ~2 mag fainter and has line velocities ~5000 km/s smaller (near maximum brightness) than a normal SN Ia. SN 2008ha had a remarkably short rise time of only ~10 days, significantly shorter than either SN 2002cx-like objects (~15 days) or normal SNe Ia (~19.5 days). The bolometric light curve of SN 2008ha indicates that SN 2008ha peaked at L_peak = (9.5 +/- 1.4) x 10^40 ergs/s, making SN 2008ha perhaps the least luminous SN ever observed. From its peak luminosity and rise time, we infer that SN 2008ha generated (3.0 +/- 0.9) x 10^-3 M_sun of 56Ni, had a kinetic energy of ~2 x 10^48 ergs, and ejected 0.15 M_sun of material. We classify three new (and one potential) members of the SN 2002cx-like class, expanding the sample to 14 (and one potential) members. The host-galaxy morphology distribution of the class is consistent with that of SNe Ia, Ib, Ic, and II. Several models for generating low-luminosity SNe can explain the observations of SN 2008ha; however, if a single model is to describe all SN 2002cx-like objects, either electron capture in Ne-Mg white dwarfs causing a core collapse, or deflagration of C-O white dwarfs with SN 2008ha being a partial deflagration and not unbinding the progenitor star, are preferred. Abridged.
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