We classified the reddest (r-J>2.2) stars observed by the NASA Kepler mission into main-sequence dwarf or evolved giant stars and determined the properties of 4216 M dwarfs based on a comparison of available photometry with that of nearby calibrator stars, as well as available proper motions and spectra. We revised the properties of candidate transiting planets using the stellar parameters, high-resolution imaging to identify companion stars, and, in the case of binaries, fitting light curves to identify the likely planet host. In 49 of 54 systems, we validated the primary as the host star. We inferred the intrinsic distribution of M dwarf planets using the method of iterative Monte Carlo simulation. We compared several models of planet orbital geometry and clustering and found that one where planets are exponentially distributed and almost precisely coplanar best describes the distribution of multiplanet systems. We determined that Kepler M dwarfs host an average of 2.2+/-0.3 planets with radii of 1-4R_{Earth}_ and orbital periods of 1.5-180d. The radius distribution peaks at ~1.2R_{Earth}_ and is essentially zero at 4R_{Earth}_, although we identify three giant planet candidates other than the previously confirmed Kepler-45b. There is suggestive but not significant evidence that the radius distribution varies with orbital period. The distribution with logarithmic orbital period is flat except for a decline for orbits less than a few days. 12 candidate planets, including two Jupiter-size objects, experience an irradiance below the threshold level for a runaway greenhouse on an Earth-like planet and are thus in a 'habitable zone'.