We have analyzed data from two sets of calibration observations of the Moon made by the Chandra X-Ray Observatory. In addition to obtaining a spectrum of the bright side that shows several distinct fluorescence lines, we also clearly detect time-variable soft X-ray emission, primarily O VII Kα and O VIII Lyα, when viewing the optically dark side. The apparent dark-side brightness varied in time by at least an order of magnitude, up to ~2 × 10^-6 photons s^-1 arcmin^-2 cm^-2 between 500 and 900 eV, which is comparable to the typical keV-band background emission measured in the ROSAT All-Sky Survey. The spectrum is also very similar to background spectra recorded by Chandra in low- or moderate-brightness regions of the sky. Over a decade ago, ROSAT also detected soft X-rays from the dark side of the Moon, which were tentatively ascribed to continuum emission from energetic solar wind electrons impacting the lunar surface. The Chandra observations, however, with their better spectral resolution, combined with contemporaneous measurements of solar wind parameters, strongly favor charge transfer between highly charged solar wind ions and neutral hydrogen in the Earth's geocorona as the mechanism for this emission. We present a theoretical model of geocoronal emission and show that predicted spectra and intensities match the Chandra observations very well. We also model the closely related process of heliospheric charge transfer and estimate that the total charge transfer flux observed from Earth amounts to a significant fraction of the soft X-ray background, particularly in the ROSAT keV band.