The Chandra Orion Ultradeep Project ( COUP ) provides the most comprehensive dataset ever acquired on the X-ray emission of pre-main sequence stars . In this paper , we study the nearly 600 X-ray sources that can be reliably identified with optically well characterized T Tauri stars ( TTS ) in the Orion Nebula Cluster . With a detection limit of L _ { X,min } \sim 10 ^ { 27.3 } erg/sec for lightly absorbed sources , we detect X-ray emission from more than 97 % of the optically visible late-type ( spectral types F to M ) cluster stars . This proofs that there is no “ X-ray quiet ” population of late-type stars with suppressed magnetic activity . We use this exceptional optical , infrared , and X-ray data set to study the dependencies of the X-ray properties on other stellar parameters . All TTS with known rotation periods lie in the saturated or super-saturated regime of the relation between activity and Rossby numbers seen for main-sequence ( MS ) stars , but the TTS show a much larger scatter in X-ray activity than seen for the MS stars . Strong near-linear relations between X-ray luminosities , bolometric luminosities and mass are present . We also find that the fractional X-ray luminosity L _ { X } / L _ { bol } rises slowly with mass over the 0.1 - 2 M _ { \odot } range . The plasma temperatures determined from the X-ray spectra of the TTS are much hotter than in MS stars , but seem to follow a general solar-stellar correlation between plasma temperature and activity level . The scatter about the relations between X-ray activity and stellar parameters is larger than the expected effects of X-ray variability , uncertainties in the variables , and unresolved binaries . This large scatter seems to be related to the influence of accretion on the X-ray emission . While the X-ray activity of the non-accreting TTS is consistent with that of rapidly rotating MS stars , the accreting stars are less X-ray active ( by a factor of \sim 2 - 3 on average ) and produce much less well defined correlations than the non-accretors . We discuss possible reasons for the suppression of X-ray emission by accretion and the implications of our findings on long-standing questions related to the origin of the X-ray emission from young stars , considering in particular the location of the X-ray emitting structures and inferences for pre-main-sequence magnetic dynamos .