  The eROSITA Final Equatorial-Depth Survey has been carried out during the PV
phase of the SRG/eROSITA telescope and completed in November 2019. This survey
is designed to provide the first eROSITA-selected sample of galaxy clusters and
groups and to test the predictions for the all-sky survey in the context of
cosmological studies with clusters. In the 140 deg$^2$ area covered by eFEDS,
542 candidate clusters and groups are detected as extended X-ray sources, down
to a flux of $\sim10^{-14} $erg/s/cm$^2$ in the soft band (0.5-2 keV) within
1'. In this work, we provide the catalog of candidate galaxy clusters and
groups in eFEDS. We perform imaging and spectral analysis on the eFEDS clusters
with eROSITA X-ray data, and study the properties of the sample. The clusters
are distributed in the redshift range [0.01, 1.3], with the median redshift at
0.35. We obtain the ICM temperature measurement with $>2\sigma$ c.l. for
$\sim$1/5 (102/542) of the sample. The average temperature of these clusters is
$\sim$2 keV. Radial profiles of flux, luminosity, electron density, and gas
mass are measured from the precise modeling of the imaging data. The selection
function, the purity and completeness of the catalog are examined and discussed
in detail. The contamination fraction is $\sim1/5$ in this sample, dominated by
misidentified point sources. The X-ray Luminosity Function of the clusters
agrees well with the results obtained from other recent X-ray surveys. We also
find 19 supercluster candidates in eFEDS, most of which are located at
redshifts between 0.1 and 0.5. The eFEDS cluster and group catalog provides a
benchmark proof-of-concept for the eROSITA All-Sky Survey extended source
detection and characterization. We confirm the excellent performance of eROSITA
for cluster science and expect no significant deviations from our pre-launch
expectations for the final All-Sky Survey.
