We present the results of two Chandra observations (separated by 1 yr) of the population of X-ray sources in the spiral galaxy M51 (NGC 5194 and NGC 5195). One hundred and thirteen X-ray sources have been detected in an 8A4 Â 8A4 (20:4 Â 20:4 kpc) region, and 84 and 12 of them project within the disks of NGC 5194 and NGC 5195, respectively. Nine and 28 sources have luminosities exceeding 1 Â 10 39 ergs s À1 (ultraluminous X-ray sources or ULXs) and 1 Â 10 38 ergs s À1 in the 0.5-8 keV band, respectively, assuming that 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 Lð0:5 8 keVÞ > 10 38 ergs s À1 is well described by a power law, N ½>L(0.5-8 keVÞ / L(0.5-8 keV) À with ¼ 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 an MCD model in two ULXs. One ULX (NGC 5194 source 69) shows drastic spectral steepening accompanied by a decline in luminosity by a factor of 3460 in the 2-10 keV band between the two observations. This source also exhibited a possible period of 2.1 hr in the 2000 observation. Another ULX (NGC 5194 source 26) shows strong emission lines from highly ionized species. 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 an MCD interpretation, their MCD parameters imply that most of the X-ray sources are stellar mass ($5-10 M ) black holes accreting near or above the Eddington rate, although other possibilities (intermediate-mass black holes and relativistically beamed emission) cannot be excluded. The power-law sources may instead represent Comptonized disk or nonthermal emission. Two ULXs have very soft spectra; MCD models require kT % 0:1 keV. We discuss the possibility that this soft emission originates in an accretion disk around an intermediate-mass black hole. We also present a study of the nucleus of, and discrete sources (including two ULXs) in, the companion galaxy NGC 5195.