The diffusive shock acceleration process can accelerate particles to a maximum energy depending on the shock speed and age and on any competing loss processes on the particles. The shock waves of young supernova remnants can easily accelerate electrons to energies in excess of 1 TeV, where they can produce X-rays by the synchrotron process. I describe a detailed calculation of the morphology and spectrum of synchrotron X-rays from supernova remnants. Remnants are assumed to be spherical and in the Sedov evolutionary phase, though the results are insensitive to the detailed dynamics. The upstream magnetic field is assumed uniform; downstream it is assumed to be compressed but not additionally turbulently amplified. In all cases, spectra begin to depart from power laws somewhere in the optical to UV range and roll off smoothly through the X-ray band. I show that simple approximations for the electron emissivity are not adequate; a full convolution of the individual electron synchrotron emissivity with a calculated electron distribution at each point in the remnant is required. Models limited by the finite shock age, by synchrotron or inverse Compton losses on electrons, or by escape of electrons above some energy have characteristically different spectral shapes, but within each class, models resemble one another strongly and can be related by simple scalings. The images and spectra depend primarily on the remnant age, the upstream magnetic field strength, and the level of magnetic turbulence near the shock in which the electrons scatter. In addition, images depend on the viewing or aspect angle between the upstream magnetic field and the line of sight. The diffusion coefficient is assumed to be proportional to particle energy (or mean free path proportional to gyroradius), but I investigate the possibility that the proportionality constant becomes much larger above some energy, corresponding to an absence of long-wavelength MHD waves. Models producing similar spectra may differ significantly in morphology, which allows for possible discriminations. I parameterize the model spectra in terms of a slope at 4 keV and a factor by which the X-ray flux density at that energy falls below the extrapolated radio spectrum. Synchrotron radiation may contribute significantly to the X-ray emission of remnants up to several thousand years old.