XENON10 is the first two-phase xenon time projection chamber (TPC) developed within the XENON dark matter search program. The TPC, with an active liquid xenon (LXe) mass of about 14 kg, was installed at the Gran Sasso Underground Laboratory (LNGS) in Italy, and operated for more than one year, with excellent stability and performance. Results from a dark matter search with XENON10 have been published elsewhere. In this paper, we summarize the design and performance of the detector and its subsystems, based on calibration data using sources of gamma-rays and neutrons as well as background and Monte Carlo simulation data. The results on the detector's energy threshold, position resolution, and overall efficiency show a performance that exceeds design specifications, in view of the very low energy threshold achieved (<10 keVr) and low background rate achieved.