We used the High-Energy Transmission Grating Spectrometer on board the Chandra X-Ray Observatory to perform two observations, separated by 3 weeks, of the Orion Trapezium region. The zeroth-order images on the Advanced CCD Imaging Spectrometer provide spatial resolution of and 0.5" moderate energy resolution. Within a 160" x 140" region around the Orion Trapezium, we resolve 111 X-ray sources with luminosities between 7 x 10(28) and 2 x 10(32) ergs s(-1). We do not detect any diffuse emission. All but six sources are identified. From spectral fits of the three brightest stars in the Trapezium, we determine the line-of-sight column density to be N-H = (1/93 +/- 0.29) x 10(21) cm(-2). Many sources appear much more heavily absorbed, with N-H in the range of 10(22)-10(23) cm(-2). A large fraction of sources also show excursions in luminosity by more than a factor 5 on timescales greater than 50 ks; many are detected only in one of the observations. The main objective of this paper is to study the Orion Trapezium and its close vicinity. All five Trapezium stars are bright in X-rays, with theta (1) Ori C accounting for about 60% of the total luminosity of the Trapezium. The CCD spectra of the three very early-type members can be fitted with a two-temperature thermal spectrum with a soft component of k(T) similar to 0.8 keV and a hard component of k(T) similar to 2-3 keV. theta (1) Ori B is an order of magnitude fainter than theta (1) Ori E and shows only a hard spectrum of k(T) similar to 3 keV. theta (1) Ori D is another order of magnitude fainter than theta (1) Ori B, with only a kT similar to 0.7 keV component. We discuss these results in the context of stellar wind models. We detect eight additional, mostly variable X-ray sources in the close vicinity of the Trapezium. They are identified with thermal and nonthermal radio sources, as well as infrared and optical stars. Five of these X-ray sources are identified with proplyds, and we argue that the X-ray emission originates from class I, II, and III protostars at the cores of the proplyds.