© 2018 American Physical Society. 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.