We report here the results of the first Chandra X-Ray Observatory observations of the globular cluster M28 (NGC 6626). 46 X-ray sources are detected, of which 12 lie within one core radius of the center. We show that the apparently extended X-ray core emission seen with the ROSAT HRI is due to the superposition of multiple discrete sources for which we determine the X-ray luminosity function down to a limit of about 6xE30 erg/s. For the first time the unconfused phase-averaged X-ray spectrum of the 3.05-ms pulsar B1821--24 is measured and found to be best described by a power law with photon index ~ 1.2. Marginal evidence of an emission line centered at 3.3 keV in the pulsar spectrum is found, which could be interpreted as cyclotron emission from a corona above the pulsar's polar cap if the the magnetic field is strongly different from a centered dipole. The unabsorbed pulsar flux in the 0.5--8.0 keV band is ~3.5xE-13 ergs/s/cm^2. Spectral analysis of the 5 brightest unidentified sources is presented. Based on the spectral parameters of the brightest of these sources, we suggest that it is a transiently accreting neutron star in a low-mass X-ray binary, in quiescence. Fitting its spectrum with a hydrogen neutron star atmosphere model yields the effective temperature T_eff^∞ = 90^{+30}_{-10} eV and the radius R_NS^∞ = 14.5^{+6.9}_{-3.8} km. In addition to the resolved sources, we detect fainter, unresolved X-ray emission from the central core of M28. Using the Chandra-derived positions, we also report on the result of searching archival Hubble Space Telescope data for possible optical counterparts. Comment: Accepted for publication in ApJ; 22 pages, 8 figures, 5 tables