We present JWST observations of the Crab Nebula, the iconic remnant of the historical SN 1054. The observations include NIRCam and MIRI imaging mosaics plus MIRI/MRS spectra that probe two select locations within the ejecta filaments. We derive a high-resolution map of dust emission and show that the grains are concentrated in the innermost, high-density filaments. These dense filaments coincide with multiple synchrotron bays around the periphery of the Crab's pulsar wind nebula (PWN). We measure synchrotron spectral index changes in small-scale features within the PWN’s torus region, including the well-known knot and wisp structures. The index variations are consistent with Doppler boosting of emission from particles with a broken power-law distribution, providing the first direct evidence that the curvature in the particle injection spectrum is tied to the acceleration mechanism at the termination shock. We detect multiple nickel and iron lines in the ejecta filaments and use photoionization models to derive nickel-to-iron abundance ratios that are a factor of 3-8 higher than the solar ratio. We also find that the previously reported order-of-magnitude higher Ni/Fe values from optical data are consistent with the lower values from JWST when we reanalyze the optical emission using updated atomic data and account for local extinction from dust. We discuss the implications of our results for understanding the nature of the explosion that produced the Crab Nebula and conclude that the observational properties are most consistent with a low-mass Fe core-collapse supernova, even though an electron-capture explosion cannot be ruled out.