Using high-quality optical images from the Carnegie-Irvine Galaxy Survey, we perform multicomponent decompositions of S0 galaxies (S0s) to derive accurate structural parameters to constrain their physical origin. Many S0s do not host prominent bulges. S0s have a broad distribution of bulge-to-total ratios (B/T) and Sersic indices (n), with average values of B/T=0.34+/-0.15 and n=2.62+/-1.02, qualitatively consistent with the notion that S0s define a parallel sequence with and may have evolved from spiral galaxies. This is further reinforced by the incidence of bars and lenses in S0s, which, when compared with the statistics in spirals, supports the idea that lenses are demised bars. However, despite their wide range of prominence, the bulges of S0s form a surprisingly uniform sequence on both the Kormendy and fundamental plane relations. There is no evidence for population dichotomy or other indications of differentiation into classical and pseudo bulges. Most of the S0s reside in the field and in groups; cluster environment is not a necessary condition for S0 production. The properties of S0 bulges show little correlation with environmental indicators, after the dependence of galaxy stellar mass on environment is taken into account. As the bulges of late-type spirals and S0s are intrinsically different, and environmental effects that may account for such evolution appear to be minimal, we conclude that late- type spirals are not plausible progenitors of S0s. The bulges of S0s likely formed at an early epoch, after which secular processes contributed little to their subsequent evolution.