The OLYMPUS experiment was designed to measure the ratio between the positron-proton and electron-proton elastic scattering cross-sections, with the goal of determining the contribution of two-photon exchange to the elastic cross-section. Two-photon exchange might resolve the discrepancy between measurements of the proton form factor ratio, μpGEp/GMp, made using polarization techniques and those made in unpolarized experiments. OLYMPUS operated on the DORIS storage ring at DESY, alternating between 2.01 GeV electron and positron beams incident on an internal hydrogen gas target. The experiment used a toroidal magnetic spectrometer instrumented with drift chambers and time-of-flight detectors to measure rates for elastic scattering over the polar angular range of approximately 25°-75°. Symmetric Møller/Bhabha calorimeters at 1.29° and telescopes of GEM and MWPC detectors at 12° served as luminosity monitors. A total luminosity of approximately 4.5 fb-1 was collected over two running periods in 2012. This paper provides details on the accelerator, target, detectors, and operation of the experiment.