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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 $\rho_11^\circ$ 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., Comment: 42 figures, 4 tables, v3 matches accepted version that will be published in PASJ (minor changes from v2). For high-resolution figures and cross-references with other HSC articles that will be in the same PASJ issue, please see the published version of the article