We have measured the central structural properties for a sample of S0-Sbc galaxies down to scales of ~10 pc using Hubble Space Telescope NICMOS images. Central star clusters are found in 58% of our sample. Their near-infrared luminosities scale with the host bulge luminosities, as reported in 2003 by Balcells and coworkers. In terms of photometric masses, the relation is MPS=107.75+/-0.15 (Mbul/1010Msolar)0.76+/-0.13. Put together with recent data on bulges hosting supermassive black holes, we infer a nonlinear dependency of the ``central massive object'' mass on the host bulge mass such that MCMO/Msolar=107.51+/-0.06(Mbul/1010 Msolar)0.84+/-0.06. The linear relation presented by Ferrarese and coworkers may be biased at the low-mass end by the inclusion of the disk light from cluster lenticular galaxies. Bulge-disk decompositions reaching to the outer disk show that ~90% of our galaxies possess central light excesses that can be modeled with an inner exponential and/or an unresolved source. All the extended nuclear components, with sizes of a few hundred parsecs, have disky isophotes, which suggest that they may be inner disks, rings, or bars; their colors are redder than those of the underlying bulge, arguing against a recent origin for their stellar populations. Surface brightness profiles (of the total galaxy light, and the bulge component on its own) rise inward to the resolution limit of the data, with a continuous distribution of logarithmic slopes from the low values typical of dwarf ellipticals (0.1<=γ<=0.3) to the high values (γ~1) typical of intermediate-luminosity ellipticals; the nuclear slope bimodality reported by others is not present in our sample. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. Based on observations made with the Isaac Newton Telescope operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.