We have investigated the X-ray spectral properties of a collection of low-mass X-ray binaries (LMXBs) within a sample of 15 nearby early-type galaxies using proprietary and archival data from the Chandra X-Ray Observatory. We find that the spectrum of the sum of the sources in a given galaxy is remarkably similar from galaxy to galaxy when only sources with X-ray luminosities less than 10(39) ergs s(-1) (0.3-10 keV) are considered. Fitting these lower luminosity sources in all galaxies simultaneously with a power-law model led to a best-fit power-law exponent of Gamma = 1.56 +/- 0.02 (90% confidence), and using a thermal bremsstrahlung model yielded kT(brem) = 7.3 +/- 0.3 keV. This is the tightest constraint to date on the spectral properties of LMXBs in external galaxies. The spectral properties of the LMXBs do not vary with galactic radius out to three effective radii. There is also no apparent difference between the spectral properties of LMXBs that reside within globular clusters and those that do not. We demonstrate how the uniformity of the spectral properties of LMXBs can lead to more accurate determinations of the temperature and metallicity of the hot gas in galaxies that have comparable amounts of X-ray emission from hot gas and LMXBs. Although few in number in any given galaxy, sources with luminosities of (1-2) x 10(39) ergs s(-1) are present in 10 of the galaxies. The spectra of these luminous sources are softer than the spectra of the rest of the sources and are consistent with the spectra of Galactic black hole X-ray binary candidates when they are in their very high state. The spatial distribution of these sources is much flatter than the optical light distribution, suggesting that a significant portion of them must reside within globular clusters. The simplest explanation of these sources is that they are similar to10-15 M. black holes accreting near their Eddington limit. The spectra of these sources are very different from those of ultraluminous X-ray sources (ULXs) that have been found within spiral galaxies, suggesting that the two populations of X-ray-luminous objects have different formation mechanisms. The number of sources with apparent luminosities above 2 x 10(39) ergs s(-1) when determined using the distance of the galaxy is equal to the number of expected background active galactic nuclei and thus appears not to be associated with the galaxy, indicating that very luminous sources are absent or very rare in early-type galaxies. The lack of ULXs within elliptical galaxies strengthens the argument that ULXs are associated with recent star formation.