We discuss a series of spectra of the Type Ic (helium-poor Ib) supernova (SN) 1987M in the Sc galaxy NGC 2715, obtained over a 6 month interval following its discovery. Two spectra of SNe Ia, SN 1987L in NGC 2336 and SN 1987N in NGC 7606, are also illustrated for comparison. Near maximum brightness, SN 1987M showed all the spectroscopic characteristics of the "helium-poor" variety of SNe Ib identified in 1987 by Wheeler and collaborators, but it may have been subluminous by only 0.6 mag (B) instead of the average value of 1.5 mag. The possible presence of very weak Hα must be verified through spectral synthesis. Strong P Cygni profiles of O I λ 7774 and Ca II are visible at early times in the near-infrared spectrum, as previously predicted. The Type IIb SN 1987K, by contrast, exhibits prominent Hα but no O I λ7774, although the two objects appear quite similar at blue and visual wavelengths. Strong Co II and Fe II absorption may be present at near-ultraviolet wavelengths in SN 1987M, as in SNe Ia. The data clearly reveal, for the first time, the remarkable transition to the "supernebular phase" of SNe Ib and Ic. Unambiguous [O I] and [Ca II] lines begin to appear less than two months past maximum. The ejected mass is estimated to be small, M ≲ 1 M⊙, although this number is highly uncertain. Multicolor light curves of SN 1987M, derived from CCD images and spectra, show a rapid decline from maximum brightness, perhaps even steeper than that of typical SNe I. This also suggests a relatively low ejected mass. Furthermore, unlike many SNe Ib, SN 1987M was not superposed on a luminous H II region. It is possible that SNe Ic, such as SN 1987M, have different progenitors or explosion mechanisms than SNe Ib. Specifically, we postulate that SNe Ic result from explosions of white dwarfs, while core collapse in massive, hydrogen-deficient stars produces SNe Ib.