The SRG/eROSITA All-Sky Survey: The first catalog of galaxy clusters and groups in the Western Galactic Hemisphere

Clusters of galaxies can be used as powerful probes to study astrophysical processes on large scales, test theories of the growth of structure, and constrain cosmological models. The driving science goal of the SRG/eROSITA All-Sky Survey (eRASS) is to assemble a large sample of X-ray-selected clusters with a well-defined selection function to determine the evolution of the mass function and, hence, the cosmological parameters. We present here a catalog of 12247 optically confirmed galaxy groups and clusters detected in the 0.2-2.3 keV as extended X-ray sources in a 13,116deg$^2$ region in the western Galactic hemisphere of the sky, which eROSITA surveyed in its first six months of operation. The clusters in the sample span the redshift range $0.003<z<1.32$. The majority (68%) of these clusters, 8361 sources, represent new discoveries without known counterparts in the literature. The mass range of the sample covers three orders of magnitude from $5\times10^{12}M_{\rm sun}$ to $2\times10^{15}M_{\rm sun}$. We construct a sample for cosmology with a higher purity level (~95%) than the primary sample, comprising 5259 securely detected and confirmed clusters in the 12791deg$^{2}$ common footprint with the DESI Legacy Survey DR10. We characterize the X-ray properties of each cluster, including their flux, luminosity and temperature, the total mass, gas mass, gas mass fraction, and mass proxy $Y_{X}$. These are determined within two apertures, 300 kpc, and the overdensity radius $R_{500}$, and are calculated by applying a forward modeling approach with a rigorous X-ray background treatment, K-factor, and the Galactic absorption corrections. Population studies utilizing LogN-LogS, the number of clusters detected above a given flux limit, and the luminosity function show overall agreement with the previous X-ray surveys after accounting for the survey completeness and purity (ABRIDGED)