The X-ray populations of Local Group galaxies have been classified in detail by Einstein, ROSAT, and ASCA, revealing a mix of binaries, supernova remnants (SNRs), and H ii regions. However, these observatories were unable to resolve X-ray sources in galaxies beyond the Local Group. With Chandra's exquisite spatial resolution, we are able to resolve sources in a sample of nearby galaxies. We show that there are highly significant differences in the X-ray colors of sources in bulge and disk systems. In particular, we find that there is a population of X-ray-soft, faint sources in disk galaxies not seen in bulges and a smaller population of hard sources also seen preferentially in disk systems. These differences can be used as a basis to classify sources as low-and high-mass X-ray binaries, SNRs, and supersoft sources. We suggest that the soft sources seen preferentially in disks are probably dominated by SNRs, although we cannot rule out the possibility that they are a new population of absorbed, faint, supersoft accretion sources associated with the young stellar population. The hard sources seen in disks but not bulges we identify as high-mass X-ray binaries. While it is impossible to classify any individual source on the basis of X-ray color alone, the observations presented here suggest that it is possible to separate sources into groups dominated by one or two source types. This classification scheme is likely to be very useful in population studies, where it is crucial to distinguish between different classes of objects.