  We investigate the X-ray number counts in the 1-2 Ms Chandra Deep Fields
(CDFs) to determine the contributions of faint X-ray source populations to the
extragalactic X-ray background (XRB). X-ray sources were separated into Active
Galactic Nuclei (AGN), star-forming galaxies, and Galactic stars based on
several criteria. We find that AGN continue to dominate the number counts in
the 0.5-2.0 keV and 2-8 keV bands. At flux limits of ~2.5e-17 erg cm-2 s-1
(0.5-2.0 keV) and ~1.4e-16 erg cm-2 s-1 (2-8 keV), the overall AGN source
densities are 7166 (+304/-292) and 4558 (+216/-207) sources deg-2,
respectively; these are factors of ~10-20 higher than found in the deepest
optical spectroscopic surveys. While still a minority, the number counts of
star-forming galaxies climb steeply such that they eventually achieve source
densities of 1727 (+187/-169) and 711 (+270/-202) sources deg-2 at the CDF
0.5-2.0 keV and 2-8 keV flux limits, respectively. Adopting recent XRB flux
densities measurements, the CDFs resolve a total of 89.5% (+5.9%/-5.7%) and
86.9% (+6.6%/-6.3%) of the extragalactic 0.5-2.0 keV and 2-8 keV XRBs,
respectively. Extrapolation of the number-count slopes can easily account for
the entire 0.5-2.0 keV and 2-8 keV XRBs to within statistical errors. We also
revisit the reported differences between the CDF-North and CDF-South number
counts, finding that the two fields are consistent except for sources in the
2-8 keV band below F(2-8 keV)~1e-15 erg cm-2 s-1, where deviations gradually
increase to ~3.9 sigma.
