We present the observations of the gravitationally lensed system QSO 2237+0305 performed with the Chandra X-ray Observatory on 2000 Sept. 6, and on 2001 Dec. 8 for 30.3 ks and 9.5 ks, respectively. Imaging analysis resolves the four X-ray images of the Einstein Cross. A possible fifth image is detected; however, this detection less certain. Fits to the combined spectrum of all images of the Einstein Cross assuming a simple power law with Galactic and intervening absorption at the lensing galaxy yield a photon index of 1.90(+0.05,-0.05). For the first observation, this spectral model yields a 0.4-8.0 keV X-ray flux of 4.6e-13 erg cm-2 s-1 and a 0.4-8.0 keV lensed luminosity of 1.0e46 erg s-1. The source exhibits variability both over long and short time scales. The X-ray flux has dropped by 20% between the two observations, and the K-S test showed that image A is variable at the 97% confidence level within the first observation. Furthermore, a possible time-delay of 2.7(+0.5,-0.9) hours between images A and B with image A leading is detected in the first observation. The X-ray flux ratios of the images are consistent with the optical flux ratios which are affected by microlensing suggesting that the X-ray emission is also microlensed. A comparison between our measured column densities and those inferred from extinction measurements suggests a higher dust-to-gas ratio in the lensing galaxy than the average value of our Galaxy. Finally, we report the detection at the 99.99% confidence level of a broad emission feature near the redshifted energy of the Fe Kα line in only the spectrum of image A. The rest frame energy, width, and equivalent width of this feature are E = 5.7(+0.2,-0.3) keV, sigma = 0.87(+0.30,-0.15) keV, and EW = 1200(+300,-200) eV, respectively. Comment: 21 pages including 7 tables and 5 figures. Accepted for publication in ApJ