  The Serendipitous Extragalactic X-ray Source Identification (SEXSI) Program
is designed to extend greatly the sample of identified extragalactic hard X-ray
2-10 keV sources at intermediate fluxes ($\sim 10^{-13} - 10^{-15} erg/cm2/s$).
SEXSI, which studies sources selected from more than 2 deg$^2$, provides an
essential complement to the {\em Chandra} Deep Fields, which reach depths of $5
\times 10^{-16} erg/cm2/s$ (\hardrange) but over a total area of $< 0.2$
deg$^2$. In this paper we describe the characteristics of the survey and our
X-ray data analysis methodology. We present the cumulative flux distribution
for the X-ray sample of 1034 hard sources, and discuss the distribution of
spectral hardness ratios. Our lognlogs in this intermediate flux range connects
to those found in the deep fields, and by combining the data sets, we constrain
the hard X-ray population over the flux range where the differential number
counts change slope, and from which the bulk of the 2 -- 10 keV X-ray
background arises. We further investigate the lognlogs distribution separately
for soft and hard sources in our sample, finding that while a clear change in
slope is seen for the softer sample, the hardest sources are well-described by
a single power-law down to the faintest fluxes, consistent with the notion that
they lie at lower average redshift.
