We present a public catalog of transients from the Zwicky Transient Facility (ZTF) Bright Transient Survey (BTS), a magnitude-limited (m<19 mag in either the g or r filter) survey for extragalactic transients in the ZTF public stream. We introduce cuts on survey coverage, sky visibility around peak light, and other properties unconnected to the nature of the transient, and show that the resulting statistical sample is spectroscopically 97% complete at <18 mag, 93% complete at <18.5 mag, and 75% complete at <19 mag. We summarize the fundamental properties of this population, identifying distinct duration-luminosity correlations in a variety of supernova (SN) classes and associating the majority of fast optical transients with well-established spectroscopic SN types (primarily SN Ibn and II/IIb). We measure the Type Ia SN and core-collapse (CC) SN rates and luminosity functions, which show good consistency with recent work. About 7% of CC SNe explode in very low-luminosity galaxies (M_i > -16 mag), 10% in red-sequence galaxies, and 1% in massive ellipticals. We find no significant difference in the luminosity or color distributions between the host galaxies of Type II and Type Ib/c supernovae, suggesting that line-driven wind stripping does not play a major role in the loss of the hydrogen envelope from their progenitors. Future large-scale classification efforts with ZTF and other wide-area surveys will provide high-quality measurements of the rates, properties, and environments of all known types of optical transients and limits on the existence of theoretically predicted but as of yet unobserved explosions.