Planetary radii are derived for 218 exoplanets orbiting 161 M dwarf stars. Stellar radii are based on an analysis of APOGEE high-resolution near-IR spectra for a subsample of the M dwarfs; these results are used to define a stellar radius-M Ks calibration that is applied to the sample of M-dwarf planet hosts. The planetary radius distribution displays a gap over Rp ∼ 1.6─2.0 R⊕, bordered by two peaks at Rp ∼ 1.2─1.6 R⊕ (super-Earths) and 2.0─2.4 R⊕ (sub-Neptunes). The radius gap is nearly constant with exoplanetary orbital period (a power-law slope of m = +0.01−0.04+0.03 ), which is different (2σ─3σ) from m ∼ −0.10 found previously for FGK dwarfs. This flat slope agrees with pebble accretion models, which include photoevaporation and inward orbital migration. The radius gap as a function of insolation is approximately constant over the range of Sp ∼ 20─250 S⊕. The Rp─Porb plane exhibits a sub-Neptune desert for Porb < 2 days, which appears at Sp > 120 S⊕, being significantly smaller than Sp > 650 S⊕ found in the FGK planet-hosts, indicating that the appearance of the sub-Neptune desert is a function of host-star mass. Published masses for 51 exoplanets are combined with our radii to determine densities, which exhibit a gap at ρp ∼ 0.9ρ⊕, separating rocky exoplanets from sub-Neptunes. The density distribution within the sub-Neptune family itself reveals two peaks, at ρp ∼ 0.4ρ⊕ and ∼0.7ρ⊕. Comparisons to planetary models find that the low-density group are gas-rich sub-Neptunes, while the group at <ρp> ∼ 0.7ρ⊕ likely consists of volatile-rich water worlds.