Borexino has been running since May 2007 at the Laboratori Nazionali del Gran Sasso laboratory in Italy 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 Phase-I (2007–2010), Borexino first detected and then precisely measured the flux of the 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 solar neutrinos produced in the CNO cycle (carbon, nitrogen, oxigen) where carbon, nitrogen, and oxygen serve as catalysts in the fusion process. 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 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.