This dissertation presents results from a Hubble Space Telescope WFPC-2 imaging survey of the Crab Nebula in F502N ([O III]), F673N ([S II]), F631N ([O I]) and F547M (continuum). These observations have revealed several Northwest-Southeast (NW-SE) asymmetries in the morphological properties of the nebula that I attribute to the proper motion of the pulsar in the NW direction. The spin-down luminosity of the pulsar generates the pulsar wind nebula (PWN) which remains centered around the pulsar in time. The pressure of the PWN has caused a shock that propagates into freely-expanding ejecta; due to the pulsar's proper motion, the shock has propagated farther in the NW direction than it has in the SE. Assuming that density of the ejecta decreases with distance from the explosion center, I show that the shock is encountering lower density ejecta in the NW causing higher shock speeds there than in the SE. The difference in shock speeds has caused a NW-SE asymmetry in the formation of filaments and the length and development of Rayleigh-Taylor fingers. I applied an existing theoretical model of PWN to the Crab and found that the pulsar's NW motion has prevented acceleration of the filaments in the SE, explaining why the SE shows an unorganized morphology and lacks long R-T fingers. In the NW, the interface has been accelerating for at least 100 years, and the R-T instability has produced very long, well-developed fingers. I utilized the continuum image to study the correlation between dust (seen via extiction of synchrotron emission) and low-ionization gas and found that dust can be found everywhere that there is low-ionization emission; this suggests the dust formed in the supernova explosion and was swept up with the gas as the shock passed through. I have estimated the gas-to-dust ratio of several filaments to be in the range of 0.050-0.100, several times higher than what is typical for the interstellar medium.