  We present the results of two Chandra observations (separated by 1 year) of
the population of X-ray sources in the spiral galaxy M51 (NGC 5194 and NGC
5195). 113 X-ray sources have been detected in an 8'.4x8'.4 region. Nine and 28
sources have luminosities exceeding 10^39 erg/s (ULXs) and 10^38 erg/s in the
0.5-8 keV band, respectively, assuming they are associated with M51. The number
of ULXs is much higher than found in most normal spiral and elliptical
galaxies. Most of the X-ray sources and all seven of the ULXs in NGC 5194 are
located in, or close to, a spiral arm, suggesting a connection with recent star
formation. The cumulative luminosity function of the X-ray sources in NGC 5194
with Lx>10^38 erg/s is well described by a power law N(>Lx) \propto Lx^-\alpha
with \alpha=0.91. The X-ray spectra of most of the detected sources are
consistent with a power law with a photon index between 1 and 2, with a few
sources showing harder or softer spectra. The spectra of most ULXs are
consistent with both a power law and a multicolor disk blackbody (MCD) model,
while a power law model is preferable to a MCD model in two ULXs. A drastic
spectral variability and emission lines in ULXs are also presented. The masses
of the compact objects and mass accretion rates in ULXs and other X-ray sources
are not well constrained by these observations. If we adopt a MCD
interpretation, their MCD parameters imply that most of the X-ray sources are
stellar mass (~5-10 M_solar) black holes accreting near or above the Eddington
rate.(abridged)
