We present and characterize the catalog of galaxy shape measurements that will be used for cosmological weak lensing measurements in the Wide layer of the first year of the Hyper Suprime-Cam ( HSC ) survey . The catalog covers an area of 136.9 deg ^ { 2 } split into six fields , with a mean i -band seeing of 0.58 \arcsec and 5 \sigma point-source depth of i \sim 26 . Given conservative galaxy selection criteria for first year science , the depth and excellent image quality results in unweighted and weighted source number densities of 24.6 and 21.8 arcmin ^ { -2 } , respectively . We define the requirements for cosmological weak lensing science with this catalog , then focus on characterizing potential systematics in the catalog using a series of internal null tests for problems with point-spread function ( PSF ) modeling , shear estimation , and other aspects of the image processing . We find that the PSF models narrowly meet requirements for weak lensing science with this catalog , with fractional PSF model size residuals of approximately 0.003 ( requirement : 0.004 ) and the PSF model shape correlation function \rho _ { 1 } < 3 \times 10 ^ { -7 } ( requirement : 4 \times 10 ^ { -7 } ) at 0.5 ^ { \circ } scales . A variety of galaxy shape-related null tests are statistically consistent with zero , but star-galaxy shape correlations reveal additive systematics on on > 1 ^ { \circ } scales that are sufficiently large as to require mitigation in cosmic shear measurements . Finally , we discuss the dominant systematics and the planned algorithmic changes to reduce them in future data reductions .