  We present galaxy-galaxy lensing measurements from 1321 sq. deg. of the Dark
Energy Survey (DES) Year 1 (Y1) data. The lens sample consists of a selection
of 660,000 red galaxies with high-precision photometric redshifts, known as
redMaGiC, split into five tomographic bins in the redshift range $0.15 < z <
0.9$. We use two different source samples, obtained from the Metacalibration
(26 million galaxies) and Im3shape (18 million galaxies) shear estimation
codes, which are split into four photometric redshift bins in the range $0.2 <
z < 1.3$. We perform extensive testing of potential systematic effects that can
bias the galaxy-galaxy lensing signal, including those from shear estimation,
photometric redshifts, and observational properties. Covariances are obtained
from jackknife subsamples of the data and validated with a suite of log-normal
simulations. We use the shear-ratio geometric test to obtain independent
constraints on the mean of the source redshift distributions, providing
validation of those obtained from other photo-$z$ studies with the same data.
We find consistency between the galaxy bias estimates obtained from our
galaxy-galaxy lensing measurements and from galaxy clustering, therefore
showing the galaxy-matter cross-correlation coefficient $r$ to be consistent
with one, measured over the scales used for the cosmological analysis. The
results in this work present one of the three two-point correlation functions,
along with galaxy clustering and cosmic shear, used in the DES cosmological
analysis of Y1 data, and hence the methodology and the systematics tests
presented here provide a critical input for that study as well as for future
cosmological analyses in DES and other photometric galaxy surveys.
