We have obtained high-resolution (<~1") deep X-ray images of the globular cluster 47 Tucanae (NGC 104) with the Chandra X-ray Observatory to study the population of compact binaries in the high stellar density core. A 70-kilosecond exposure of the cluster reveals a centrally concentrated population of faint (L_X_~10^30-33^ergs/s) X-ray sources, with at least 108 located within the central 2'x2.5' and >~half with L_X_<~10^30.5^ergs/s. All 15 millisecond pulsars (MSPs) recently located precisely by radio observations are identified, though 2 are unresolved by Chandra. The X-ray spectral and temporal characteristics, as well as initial optical identifications with the Hubble Space Telescope, suggest that >~50 percent are MSPs, about 30 percent are accreting white dwarfs, about 15 percent are main-sequence binaries in flare outbursts, and only two to three are quiescent low-mass X-ray binaries containing neutron stars, the conventional progenitors of MSPs. An upper limit of about 470 times the mass of the sun is derived for the mass of an accreting central black hole in the cluster. These observations provide the first X-ray ``color-magnitude'' diagram for a globular cluster and census of its compact object and binary population. Observations were made on UT 16.31 - 17.22 March, 2000.