We have obtained WFPC2 images of 256 of the nearest (z ≤ 0.035) Seyfert 1, Seyfert 2, and starburst galaxies. Our 500 s broadband (F606W) exposures reveal much fine-scale structure in the centers of these galaxies, including dust lanes and patches, bars, rings, wisps, and filaments, and tidal features such as warps and tails. Most of this fine structure cannot be detected in ground-based images. We have assigned qualitative classifications for these morphological features and a Hubble type for the inner region of each galaxy, and we have also measured quantitative information such as 0″.18 and 0″.92 aperture magnitudes, position angles, and ellipticities, where possible. There is little direct evidence for unusually high rates of interaction in the Seyfert galaxies. Slightly less than 10% of all the galaxies show tidal features or multiple nuclei. The incidence of inner starburst rings is about 10% in both classes of Seyfert galaxies. In contrast, galaxies with H II region emission-line spectra appear substantially more irregular and clumpy because of their much higher rates of current star formation per unit of galactic mass. The presence of an unresolved central continuum source in our Hubble Space Telescope images is a virtually perfect indicator of a Seyfert 1 nucleus as seen by ground-based spectroscopy. Fifty-two percent of these Seyfert 1 point sources are saturated in our images; we use their wings to estimate magnitudes ranging from 15.8 to 18.5. The converse is not universally true, however, as over one-third of Seyferts with direct spectroscopic evidence for broad Balmer wings show no nuclear point source. These 34 resolved Seyfert 1's have fainter nonstellar nuclei, which appear to be more extinguished by dust absorption. Like the Seyfert 2's, they have central surface brightnesses consistent with those expected for the bulges of normal galaxies. The rates for the occurrences of bars in Seyfert 1's and 2's and non-Seyferts are the same. We found one significant morphological difference between the host galaxies of Seyfert 1 and Seyfert 2 nuclei. The Seyfert 2 galaxies are significantly more likely to show nuclear dust absorption, especially in lanes and patches that are irregular or reach close to the nucleus. A few simple tests show that the difference cannot be explained by different average redshifts or selection techniques. It is confirmed by our galaxy morphology classifications, which show that Seyfert 1 nuclei reside in earlier type galaxies than Seyfert 2 nuclei. If, as we believe, this is an intrinsic difference in host galaxy properties, it undermines one of the postulates of the strong unification hypothesis for Seyfert galaxies, that they merely appear different because of the orientation of their central engine. The excess galactic dust we see in Seyfert 2's may cause substantial absorption that obscures their hypothesized broad emission line regions and central nonstellar continua. This galactic dust could produce much of the absorption in Seyfert 2 nuclei that had instead been attributed to a thick dusty accretion torus forming the outer part of the central engine.