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We present results from an X-ray imaging survey of the young cluster NGC 2264, carried out with the European Photon Imaging Cameras (EPIC) on board the XMM-Newton spacecraft. The X-ray data are merged with extant optical and near-infrared photometry, spectral classifications, Hα emission strengths, and rotation periods to examine the interrelationships between coronal and chromospheric activity, rotation, stellar mass, and internal structure for a statistically significant sample of pre-main-sequence stars. A total of 300 distinct X-ray sources can be identified with optical or near-infrared counterparts. The sources are concentrated within three regions of the cluster: in the vicinity of S Mon, within the large emission/reflection nebulosity southwest of S Mon, and along the broad ridge of molecular gas that extends from the Cone Nebula to the NGC 2264 IRS 2 field. From the extinction-corrected color-magnitude diagram of the cluster, ages and masses for the optically identified X-ray sources are derived. A median age of ∼2.5 Myr and an apparent age dispersion of ∼5 Myr are suggested by pre-main-sequence evolutionary models. The X-ray luminosity of the detected sources appears well correlated with bolometric luminosity, although there is considerable scatter in the relationship. Stellar mass contributes significantly to this dispersion, while isochronal age and rotation do not. X-ray luminosity and mass are well correlated such that L_X∝(M/M_☉)^1.5, similar to the relationship found within the younger Orion Nebula Cluster. No strong evidence is found for a correlation between E_H–K, the near-infrared color excess, and the fractional X-ray luminosity, which suggests that optically thick dust disks have little direct influence on the observed X-ray activity levels. Among the X-ray-detected weak-line T Tauri stars, the fractional X-ray luminosity, L_X/L_bol, is moderately well correlated with the fractional Hα luminosity, L_Hα/L_bol, but only at the 2 σ level of significance. The cumulative distribution functions for the X-ray luminosities of the X-ray-detected classical and weak-line T Tauri stars within the cluster are comparable, assuming the demarcation between the two classes is at an Hα equivalent width of 10 Å. However, if the nondetections in X-rays for the entire sample of Hα emitters known within the cluster are taken into account, then the cumulative distribution functions of these two groups are clearly different, such that classical T Tauri stars are underdetected by at least a factor of 2 relative to the weak-line T Tauri stars. Examining a small subsample of X-ray-detected stars that are probable accretors based on the presence of strong Hα emission and near-infrared excess, we conclude that definitive nonaccretors are ∼1.6 times more X-ray luminous than their accreting counterparts. In agreement with earlier published findings for the Orion Nebula Cluster, we find a slight positive correlation (valid at the 2 σ confidence level) between L_X/L_bol and rotation period in NGC 2264. The lack of a strong anticorrelation between X-ray activity and rotation period in the stellar population of NGC 2264 suggests that either the deeply convective T Tauri stars are rotationally saturated or the physical mechanism responsible for generating magnetic fields in pre-main-sequence stars is distinct from the one that operates in evolved main-sequence stars.