(abridged:) The XMM-Newton Extended Survey of the Taurus Molecular Cloud (XEST) surveys the most populated ~5 square degrees of the Taurus star formation region, using the XMM-Newton X-ray observatory to study the thermal structure, variability, and long-term evolution of hot plasma, to investigate the magnetic dynamo, and to search for new potential members of the association. Many targets are also studied in the optical, and high-resolution X-ray grating spectroscopy has been obtained for selected bright sources. The X-ray spectra have been coherently analyzed with two different thermal models (2-component thermal model, and a continuous emission measure distribution model). We present overall correlations with fundamental stellar parameters that were derived from the previous literature. A few detections from Chandra observations have been added. The present overview paper introduces the project and provides the basic results from the X-ray analysis of all sources detected in the XEST survey.Comprehensive tables summarize the stellar properties of all targets surveyed. The survey goes deeper than previous X-ray surveys of Taurus by about an order of magnitude and for the first time systematically accesses very faint and strongly absorbed TMC objects. We find a detection rate of 85% and 98% for classical and weak-line T Tau stars (CTTS resp. WTTS), and identify about half of the surveyed protostars and brown dwarfs. Overall, 136 out of 169 surveyed stellar systems are detected. We describe an X-ray luminosity vs. mass correlation, discuss the distribution of X-ray-to-bolometric luminosity ratios, and show evidence for lower X-ray luminosities in CTTS compared to WTTS. Detailed analysis (e.g., variability, rotation-activity relations, influence of accretion on X-rays) will be discussed in a series of accompanying papers.