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We describe observed properties of the Type Iax class of supernovae (SNe Iax), consisting of SNe observationally similar to its prototypical member, SN 2002cx. The class currently has 25 members, and we present optical photometry and/or optical spectroscopy for most of them. SNe Iax are spectroscopically similar to SNe Ia, but have lower maximum-light velocities (2000 < |v| < 8000 km/s), typically lower peak magnitudes (-14.2 > M_V,peak > -18.9 mag), and most have hot photospheres. Relative to SNe Ia, SNe Iax have low luminosities for their light-curve shape. There is a correlation between luminosity and light-curve shape, similar to that of SNe Ia, but offset from that of SNe Ia and with larger scatter. Despite a host-galaxy morphology distribution that is highly skewed to late-type galaxies without any SNe Iax discovered in elliptical galaxies, there are several indications that the progenitor stars are white dwarfs (WDs): evidence of C/O burning in their maximum-light spectra, low ejecta masses, strong Fe lines in their late-time spectra, a lack of X-ray detections, and deep limits on massive stars and star formation at the SN sites. However, two SNe Iax show strong He lines in their spectra. The progenitor system and explosion model that best fits all of the data is a binary system of a C/O WD that accretes matter from a He star and has a significant deflagration. At least some of the time, this explosion will not disrupt the WD. We estimate that in a given volume there are 31^+17_-13 SNe Iax for every 100 SNe Ia, and for every 1 M_sun of iron generated by SNe Ia at z = 0, SNe Iax generate 0.052^+0.017_-0.014 M_sun. Being the largest class of peculiar SNe, thousands of SNe Iax will be discovered by LSST. Future detailed observations of SNe Iax should further our understanding of both their progenitor systems and explosions as well as those of SNe Ia.