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The Orion Nebula Cluster and the molecular cloud in its vicinity have been observed with the ACIS-I detector on board the Chandra X-ray Observatory with 23 hr exposure in two observations. We detect 1075 X-ray sources, most with subarcsecond positional accuracy. Ninety-one percent of the sources are spatially associated with known stellar members of the cluster, and an additional 7% are newly identified deeply embedded cloud members. This provides the largest X-ray study of a pre-main-sequence stellar population and covers the initial mass function from brown dwarfs up to a 45 M☉ O star. Source luminosities span 5 orders of magnitude from log L_X ≃ 28.0 to 33.3 ergs s^(-1) in the 0.5-8 keV band, plasma energies range from 0.2 to >10 keV, and absorption ranges from log N_H < 20.0 to ~23.5 cm^(-2). Comprehensive tables providing X-ray and stellar characteristics are provided electronically. \n \nWe examine here the X-ray properties of Orion young stars as a function of mass; other studies of astrophysical interest will appear in companion papers. Results include: (a) the discovery of rapid variability in the O9.5 31 M☉ star θ^2A Ori, and several early B stars, inconsistent with the standard model of X-ray production in small shocks distributed throughout the radiatively accelerated wind; (b) support for the hypothesis that intermediate-mass mid-B through A type stars do not themselves produce significant X-ray emission; (c) confirmation that low-mass G through M type T Tauri stars exhibit powerful flaring but typically at luminosities considerably below the "saturation" level; (d) confirmation that the presence or absence of a circumstellar disk has no discernable effect on X-ray emission; (e) evidence that T Tauri plasma temperatures are often very high with T ≥ 100 MK, even when luminosities are modest and flaring is not evident; and (f) detection of the largest sample of pre-main-sequence very low-mass objects showing flaring levels similar to those seen in more massive T Tauri stars and a decline in magnetic activity as they evolve into L and T type brown dwarfs.