  Borexino has been running since May 2007 at the LNGS with the primary goal of
detecting solar neutrinos. The detector, a large, unsegmented liquid
scintillator calorimeter characterized by unprecedented low levels of intrinsic
radioactivity, is optimized for the study of the lower energy part of the
spectrum. During the Phase-I (2007-2010) Borexino first detected and then
precisely measured the flux of the 7Be solar neutrinos, ruled out any
significant day-night asymmetry of their interaction rate, made the first
direct observation of the pep neutrinos, and set the tightest upper limit on
the flux of CNO neutrinos. In this paper we discuss the signal signature and
provide a comprehensive description of the backgrounds, quantify their event
rates, describe the methods for their identification, selection or subtraction,
and describe data analysis. Key features are an extensive in situ calibration
program using radioactive sources, the detailed modeling of the detector
response, the ability to define an innermost fiducial volume with extremely low
background via software cuts, and the excellent pulse-shape discrimination
capability of the scintillator that allows particle identification. We report a
measurement of the annual modulation of the 7 Be neutrino interaction rate. The
period, the amplitude, and the phase of the observed modulation are consistent
with the solar origin of these events, and the absence of their annual
modulation is rejected with higher than 99% C.L. The physics implications of
phase-I results in the context of the neutrino oscillation physics and solar
models are presented.
