Borexino could efficiently distinguish between α and β radiation in its liquid scintillator by the characteristic time profile of its scintillation pulse. This α/β discrimination, first demonstrated on the ton scale in the counting test facility prototype, was used throughout the lifetime of the experiment between 2007 and 2021. With this method, the α events are identified and subtracted from the solar neutrino events similar to β. This is particularly important in liquid scintillators, as the α scintillation is strongly quenched. In Borexino, the prominent Po210 decay peak was a background in the energy range of electrons scattered from Be7 solar neutrinos. Optimal α/β discrimination was achieved with a multilayer perceptron neural network, with a higher ability to leverage the timing information of the scintillation photons detected by the photomultiplier tubes. An event-by-event, high efficiency, stable, and uniform pulse shape discrimination was essential in characterizing the spatial distribution of background in the detector. This benefited most Borexino measurements, including solar neutrinos in the pp chain and the first direct observation of the CNO cycle in the Sun. This paper presents key milestones in α/β discrimination in Borexino as a term of comparison for current and future large liquid scintillator detectors.