  We present a 5.4$\sigma$ detection of the pairwise kinematic
Sunyaev-Zel'dovich (kSZ) effect using Atacama Cosmology Telescope (ACT) and
$\it{Planck}$ CMB observations in combination with Luminous Red Galaxy samples
from the Sloan Digital Sky Survey (SDSS) DR15 catalog. Results are obtained
using three ACT CMB maps: co-added 150 GHz and 98 GHz maps, combining
observations from 2008-2018 (ACT DR5), which overlap with SDSS DR15 over 3,700
sq. deg., and a component-separated map using night-time only observations from
2014-2015 (ACT DR4), overlapping with SDSS DR15 over 2,089 sq. deg. Comparisons
of the results from these three maps provide consistency checks in relation to
potential frequency-dependent foreground contamination. A total of 343,647
galaxies are used as tracers to identify and locate galaxy groups and clusters
from which the kSZ signal is extracted using aperture photometry. We consider
the impact of various aperture photometry assumptions and covariance estimation
methods on the signal extraction. Theoretical predictions of the pairwise
velocities are used to obtain best-fit, mass-averaged, optical depth estimates
for each of five luminosity-selected tracer samples. A comparison of the
kSZ-derived optical depth measurements obtained here to those derived from the
thermal SZ effect for the same sample is presented in a companion paper.
