  We measure the clustering of DES Year 1 galaxies that are intended to be
combined with weak lensing samples in order to produce precise cosmological
constraints from the joint analysis of large-scale structure and lensing
correlations. Two-point correlation functions are measured for a sample of $6.6
\times 10^{5}$ luminous red galaxies selected using the \textsc{redMaGiC}
algorithm over an area of $1321$ square degrees, in the redshift range $0.15 <
z < 0.9$, split into five tomographic redshift bins. The sample has a mean
redshift uncertainty of $\sigma_{z}/(1+z) = 0.017$. We quantify and correct
spurious correlations induced by spatially variable survey properties, testing
their impact on the clustering measurements and covariance. We demonstrate the
sample's robustness by testing for stellar contamination, for potential biases
that could arise from the systematic correction, and for the consistency
between the two-point auto- and cross-correlation functions. We show that the
corrections we apply have a significant impact on the resultant measurement of
cosmological parameters, but that the results are robust against arbitrary
choices in the correction method. We find the linear galaxy bias in each
redshift bin in a fiducial cosmology to be $b(z$=$0.24)=1.40 \pm 0.08$,
$b(z$=$0.38)=1.61 \pm 0.05$, $b(z$=$0.53)=1.60 \pm 0.04$ for galaxies with
luminosities $L/L_*>$$0.5$, $b(z$=$0.68)=1.93 \pm 0.05$ for $L/L_*>$$1$ and
$b(z$=$0.83)=1.99 \pm 0.07$ for $L/L_*$$>1.5$, broadly consistent with
expectations for the redshift and luminosity dependence of the bias of red
galaxies. We show these measurements to be consistent with the linear bias
obtained from tangential shear measurements.
