The XENON1T experiment at the Laboratori Nazionali del Gran Sasso is the most sensitive direct detection experiment for dark matter in the form of weakly interacting particles (WIMPs) with masses above 6 GeV/c2 scattering off nuclei. The detector employs a dual-phase time projection chamber with 2.0 metric tons of liquid xenon in the target. A one metric ton×year exposure of science data was collected between October 2016 and February 2018. This article reports on the performance of the detector during this period and describes details of the data analysis that led to the most stringent exclusion limits on various WIMP-nucleon interaction models to date. In particular, signal reconstruction, event selection, and calibration of the detector response to nuclear and electronic recoils in XENON1T are discussed.