  We present the first study of high-precision internal proper motions (PMs) in
a large sample of globular clusters, based on Hubble Space Telescope (HST) data
obtained over the past decade with the ACS/WFC, ACS/HRC, and WFC3/UVIS
instruments. We determine PMs for over 1.3 million stars in the central regions
of 22 clusters, with a median number of ~60,000 stars per cluster. These PMs
have the potential to significantly advance our understanding of the internal
kinematics of globular clusters by extending past line-of-sight (LOS) velocity
measurements to two- or three-dimensional velocities, lower stellar masses, and
larger sample sizes. We describe the reduction pipeline that we developed to
derive homogeneous PMs from the very heterogeneous archival data. We
demonstrate the quality of the measurements through extensive Monte-Carlo
simulations. We also discuss the PM errors introduced by various systematic
effects, and the techniques that we have developed to correct or remove them to
the extent possible. We provide in electronic form the catalog for NGC 7078 (M
15), which consists of 77,837 stars in the central 2.4 arcmin. We validate the
catalog by comparison with existing PM measurements and LOS velocities, and use
it to study the dependence of the velocity dispersion on radius, stellar
magnitude (or mass) along the main sequence, and direction in the plane of the
sky (radial/tangential). Subsequent papers in this series will explore a range
of applications in globular-cluster science, and will also present the PM
catalogs for the other sample clusters.
