We have used HST imaging of the central regions of the globular cluster 47 Tucanae (=NGC 104), taken with the WFPC2 and ACS instruments between 1995 and 2002, to derive proper motions and U- and V-band magnitudes for 14,366 stars within 100'' (about 5 core radii) of the cluster center. This represents the largest set of member velocities collected for any globular cluster. The stars involved range in brightness from just fainter than the horizontal branch of the cluster to more than 2.5 mag below the main-sequence turnoff. In the course of obtaining these kinematic data, we also use a recent set of ACS images to define a list of astrometrically calibrated positions (and F475W magnitudes) for nearly 130,000 stars in a larger, 3' × 3' central area. We describe our data reduction procedures in some detail and provide the full position, photometric, and velocity data in the form of downloadable electronic tables. We have used the star counts to obtain a new estimate for the position of the cluster center and to define the density profile of main-sequence turnoff and giant branch stars in to essentially zero radius, thus constraining the global spatial structure of the cluster better than before. A single-mass, isotropic King model fit to it is then used as a rough point of reference against which to compare the gross characteristics of our proper-motion data. We search in particular for any evidence of very fast-moving stars, in significantly greater numbers than expected for the extreme tails of the velocity distribution in a sample of our size. We find that likely fewer than 0.1%, and no more than about 0.3%, of stars with measured proper motions have total speeds above the nominal central escape velocity of the cluster. At lower speeds, the proper-motion velocity distribution very closely matches that of a regular King model (which is itself nearly Gaussian given the high stellar density) at all observed radii. Considerations of only the velocity dispersion then lead to a number of results: (1) Blue stragglers in the core of 47 Tuc have a velocity dispersion σ_μ smaller than that of the cluster giants by a factor of , consistent with the former being on average twice as massive as normal, main-sequence turnoff stars. (2) The velocity distribution in the inner 5 core radii of the cluster is essentially isotropic, and the detailed dependence of σ_μ on R for the brighter stars suggests that heavy remnants contribute only a fraction of a percent to the total cluster mass. Both of these results are in keeping with earlier, more realistic multimass and anisotropic models of 47 Tuc. (3) Using a sample of 419 line-of-sight velocities measured for bright giants within R ≤ 105'', we obtain a kinematic distance to the cluster, D = 4.0 ± 0.35 kpc, formally some 10%-20% lower than recent estimates based on standard color-magnitude diagram fitting, and more consistent with the value implied by fitting to the white dwarf cooling sequence. And (4) by fitting simple models of isotropic, single-mass stellar clusters with central point masses to our observed σ_μ(R) profile, we infer a 1 σ upper limit of M_• 1000-1500 M for any intermediate-mass black hole in 47 Tuc. The formal best-fit hole mass ranges from zero, if only the kinematics of stars near the main-sequence turnoff mass are modeled, to ~700-800 M, if fainter, less massive stars are also used. We can neither confirm nor refute the hypothesis that 47 Tuc might lie on an extension of the M_•-σ relation observed for galaxy bulges.