We have spatially and spectrally resolved the sources of X-ray emission from the X-ray faint S0 galaxy NGC 1553 using an observation from the Chandra X-ray Observatory. The majority (70%) of the emission in the 0.3 - 10.0 keV band is diffuse, and the remaining 30% is resolved into 49 discrete sources. Most of the discrete sources associated with the galaxy appear to be low mass X-ray binaries (LMXBs). The luminosity function of the LMXB sources is well-fit by a broken power-law with a break luminosity comparable to the Eddington luminosity for a 1.4 solar mass neutron star. It is likely that those sources with luminosities above the break are accreting black holes and those below are mostly neutron stars in binary systems. Spectra were extracted for the total emission, diffuse emission, and sum of the resolved sources; the spectral fits for all require a model including both a soft and hard component. The diffuse emission is predominately soft while the emission from the sources is mostly hard. Approximately 24% of the diffuse emission arises from unresolved LMXBs, with the remainder resulting from thermal emission from hot gas. There is a very bright source at the projected position of the nucleus of the galaxy. The spectrum and luminosity derived from this central source are consistent with it being an AGN; the galaxy also is a weak radio source. Finally, the diffuse emission exhibits significant substructure with an intriguing spiral feature passing through the center of the galaxy. The X-ray spectrum and surface brightness of the spiral feature are consistent with adiabatic or shock compression of ambient gas, but not with cooling. This feature may be due to compression of the hot interstellar gas by radio lobes or jets associated with the AGN.
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