New and existing K-band spectra for 19 Galactic center late-type stars have been analyzed along with representative spectra of disk and bulge M giants and supergiants. Absorption strengths for strong atomic and molecular features have been measured. The Galactic center stars generally exhibit stronger absorption features centered near Na I (2.206 micron) and Ca I (2.264 micron) than representative disk M stars at the same CO absorption strength. Based on the absolute K-band magnitudes and CO and H2O absorption strengths for the Galactic center stars and known M supergiants and asymptotic giant branch (AGB) stars, we conclude that only IRS 7 must be a supergiant. Two other bright stars in our Galactic center sample are likely supergiants as well. The remaining bright, cool stars in the Galactic center that we have observed are most consistent with being intermediate mass/age AGB stars. We identify four of the Galactic center stars as long period variables based on their K-band spectral properties and associated photometric variability. Estimates of initial masses and ages for the GC stars suggest multiple epochs of star formation have occurred in the Galactic center over the last 7-100 Myr.