We fitted the surface-brightness profiles of 21 elliptical galaxies using both the Sérsic function and a new empirical model that combines an inner power law with an outer Sérsic function. The profiles are combinations of deconvolved Hubble Space Telescope (HST) profiles from the literature and ellipse fits to the full WFPC2 mosaic images and thus span a radial range from ~0.02" to about twice the half-light radius. We are able to accurately fit the entire profiles using either the Sérsic function or our new model. In doing so, we demonstrate that most, if not all, so-called ``power-law'' galaxies are better described as ``Sérsic galaxies''-they are well modeled by the three-parameter Sérsic profile into the limits of HST resolution-and that ``core'' galaxies are best understood as consisting of an outer Sérsic profile with an inner power-law cusp, which is a downward deviation from the inward extrapolation of the Sérsic profile. This definition of cores resolves ambiguities that result when the popular ``Nuker law'' is fitted to the profiles of ellipticals and bulges, particularly at lower luminosities. We also find that using the Nuker law to model core-galaxy nuclear profiles systematically overestimates the core radii by factors of 1.5-4.5 and underestimates the inner power-law slope by ~20%-40% or more.