Narrowband images of the Crab Nebula taken with the Hubble Space Telescope (HST) WFPC2 show the morphology and ionization structure of the filaments in great detail. At HST resolution, low- and high-ionization emission from filaments in the Crab differ in two complementary respects. First, low-ionization emission is found to be concentrated in very sharp structures, while high-ionization emission is predominantly found in a much more diffuse component. For example, approximately 80% of emission from [O I] λ6300 arises in features with scales of less than 0.5", while only 10% of [O III] λ5007 emission arises in such compact structures. Second, individual filaments are found to lie along a sequence of ionization structures, ranging from features in which all lines are concentrated in the same compact volume through features with low-ionization cores surrounded by high-ionization envelopes. Hester and coworkers proposed in 1996 that this sequence can be understood as the result of the nonlinear development of magnetic Rayleigh-Taylor (R-T) instabilities along the interface between the Crab synchrotron nebula and swept-up ejecta.We present photoionization models of cylindrically symmetrical filaments consisting of a quadratic core surrounded by an extended envelope. A good deal of the observed variation in filament structure in the Crab can be matched by varying the assumed density profiles in these models. This implies that variations in the development of R-T instabilities in the Crab account for much of the spectral variation within the remnant. We also present a photoionization model of a uniform, low-density medium, which reasonably matches the extended diffuse component that dominates the high-ionization emission. This envelope model produces strong [O III] but virtually no [O I]. While the He I/Hβ ratio remains fairly constant throughout a range of filament models, this ratio is a factor of 5 lower in the envelope model. We find that the apertures used in ground-based spectroscopy of the Crab generally include emission from several discrete filaments as well as a component of diffuse emission. This places a fundamental limit on what can be inferred reliably from comparison of spectra with one-dimensional photoionization models. Many filament cores coincide with dust extinction features seen in a continuum image of the Crab. We consider one such feature in detail and find that the extinction of about 1.2 mag suggests that the dust-to-gas mass ratio may be an order of magnitude higher than is typical in the interstellar medium. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract 5-26555.