We present a study of the X-ray sources detected in the vicinity of two density peaks in the Orion Molecular Cloud 1 (OMC-1) behind the Orion Nebula Cluster (ONC), as seen in the exceptionally deep (~10 days) exposure of the Chandra Orion Ultradeep Project (COUP). We focus on a 40'' × 50'' region around the Becklin-Neugebauer object and Kleinmann-Low nebula (collectively BN-KL) and a 60'' × 75'' region around OMC-1S, a secondary star-forming peak some 90'' south of BN-KL. A total of 43 and 60 X-ray sources were detected in BN-KL and OMC-1S, respectively, of which one-half and one-third, respectively, were found to be foreground members of the ONC, while the remaining sources are identified as obscured X-ray sources with column densities 22 log(N_H/cm^-2) 24. All but 5 and 18 of these sources have near-infrared stellar counterparts, and 22 of these appear to be young stellar objects (YSOs). X-ray sources are found close to four luminous mid-IR sources, namely, BN, IRc3-i2, IRc2-C, and source n; their X-ray variability and spectral properties are typical of coronal activity in low-mass stars rather than wind emission from massive stars, suggesting that the X-ray emission may be arising from companions. The X-ray light curve of the X-ray source close to BN shows a periodicity of ~8.3 days, and from an X-ray image deconvolution of the region around BN, we conclude that either BN itself or a low-mass companion with a projected separation of 200 AU was detected. On the other hand, no emission is seen from the bright radio source I, held by some to be the main source of luminosity in BN-KL. In OMC-1S, Chandra unveils a new subcluster of seven YSOs without infrared counterparts. We compare the hard-band X-ray luminosity functions of obscured X-ray sources in BN-KL and OMC-1S with unobscured X-ray sources in the ONC, and we estimate that the true population of obscured sources in BN-KL and OMC-1S is 46 and 57 stars, with 90% confidence intervals of 34-71 and 42-82 stars, respectively.