We have obtained V-band images of 45 nearby elliptical galaxies and bulges using the original Planetary Camera of the Hubble Space Telescope. The majority of the sample is at distances of 10-20 Mpc. This represents a substantial increase in the number of galaxies that have been studied at a resolution of a few parsecs. At this resolution, many galaxies reveal previously unknown central disks, dust clouds, and nuclear components. We find that galaxies have two types of brightness profiles. The first type consists of galaxies that have cores. These galaxies have brightness profiles that "break" from steep outer power laws to shallow inner cusps. The core class includes many galaxies that had cores apparently resolved from the ground. The second type consists of galaxies that have profiles that continue into the resolution limit as steep power laws, showing no evidence of cores of any sort. We thus find that all galaxies studied so far have singular brightness profiles in the sense that I(r) ~ r^-γ^ as r - 0.1", with 0 < γ < 0.3 at the few parsec scale for galaxies with cores, and γ ~ 1 for power-law galaxies. No galaxies in our sample have a central region that is constant in surface brightness. This implies that the stellar density in these systems is still increasing steeply at the HST resolution limit. Many galaxies reach stellar mass densities of ~5 x 10^4^ M_sun_ pc^-3^ at the resolution limit, appearing similar in form to M32 at radii of a few parsecs. The core and power-law profile classes correspond to the Jaffe et al. (AJ, 108, 1567 (1994)] Type I and II profiles; however, we disagree with their suggestion that the presence of a central stellar disk is closely related to, or even determines, profile type. Power-law galaxies are seen at all ellipticities, and the majority of them show no evidence for central disks.