We used the HST WFPC2 to obtain I-band images of the centers of 81 brightest cluster galaxies (BCGs), drawn from a volume-limited sample of nearby BCGs. The images show a rich variety of morphological features, including multiple or double nuclei, dust, stellar disks, point source nuclei, and central surface brightness depressions. High resolution surface brightness profiles could be inferred for 60 galaxies. Of those, 88% have well-resolved cores. The relationship between core size and galaxy luminosity for BCGs is indistinguishable from that of Faber et al. (1997, hereafter F97) for galaxies within the same luminosity range. However, the core sizes of the most luminous BCGs fall below the extrapolation of the F97 relationship rb ∼ L1.15 V . A shallower relationship rb ∼ L0.72 V fits both the BCGs and the core galaxies presented in F97. Twelve percent of the BCG sample lacks a well-resolved core; all but one of these BCGs have “power-law” profiles. Some of these galaxies have higher luminosities than any power-law galaxy identified by F97 and have physical upper limits on rb well below the values observed for core galaxies of the same luminosity. These results support the idea that the central structure of early-type galaxies is bimodal in its physical properties, but also suggest that there exist high luminosity galaxies with power-law profiles (or unusually small cores). The BCGs in the latter category tend to fall at the low end of the BCG luminosity function and tend to have low values of the quantity (the logarithmic slope of the metric luminosity as a function of radius, at 10kpc). Since theoretical calculations have shown that the luminosities and values of BCGs grow with time as a result of accretion, this suggests a scenario in which elliptical galaxies evolve from power-law profiles to core profiles through accretion and merging. This is consistent with theoretical scenarios that invoke the formation of massive black hole binaries during merger events. More generally, the prevalence of large cores in the great majority of BCGs, which are likely to have experienced several generations of galaxy merging, underscores the role of a mechanism that creates and preserves cores in such merging events. (Refer to PDF file for exact formulas).