We present results from three deep ROSAT high-resolution imager observations of the Orion Nebula star-forming region. The X-ray images contain over 1500 cataloged stars in a roughly 0.8 deg(2) region centered on the Trapezium. In all 389 distinct X-ray sources have been detected, at least two-thirds of which are associated with a single proper-motion cluster member. X-ray emission is detected from stars of all spectral types, from the massive O- and B-type components of the Trapezium to the coolest, low-mass pre-main-sequence (PMS) stars. In this paper we focus primarily on X-ray emission from the late-type PMS stars. Of the similar to 100 late-type cluster members with measured spectral types, approximately three-fourths have been detected; we have derived X-ray luminosity upper limits for the remaining stars. We found coronal X-ray emission turns on around spectral type F6, with the upper envelope of activity increasing with decreasing effective temperature. When plotted in an X-ray luminosity versus bolometric luminosity diagram, late-type PMS stars lie below a ''saturation'' line corresponding to L(X)/L(bol)similar to 10(-3). For approximately solar-mass PMS stars, we find a median X-ray luminosity similar to 1 x 10(30) ergs s(-1). The late G, K, and M stars exhibit nearly a two order of magnitude spread in X-ray luminosity and in L(X)/L(bol) at a given effective temperature. Plots of X-ray activity versus v sin i rotational velocity and rotational period appear to show no clear dependence of activity on rotation. However, because only a small fraction of late-type PMS stars in the Orion Nebula have measured v sin i or P-rot and because of uncertainties in L(X) and L(X)/L(bol), we believe the data are not conclusive on this point. Light curves of the detected X-ray sources have revealed at least 10 strong X-ray flares with characteristic rise times greater than or similar to 1 hr and decay times ranging from similar to 2 to 12 hr. All the flares have X-ray energies in excess of 3 x 10(35) ergs. Many of the X-ray sources associated with late-type cluster members have been observed in more than one ROSAT HRI exposure, allowing us to look for variability on the similar to 1 yr timescale between HRI observations. A statistical analysis of the resulting count rate ratios suggests that at least one-fourth of the late-type cluster stars are significantly variable on this timescale.
