The XENON100 dark matter experiment uses liquid xenon in a time projection chamber (TPC) to measure xenon nuclear recoils resulting from the scattering of dark matter weakly interacting massive particles (WIMPs). In this paper, we report the observation of single-electron charge signals which are not related to WIMP interactions. These signals, which show the excellent sensitivity of the detector to small charge signals, are explained as being due to the photoionization of impurities in the liquid xenon and of the metal components inside the TPC. They are used as a unique calibration source to characterize the detector. We explain how we can infer crucial parameters for the XENON100 experiment: the secondary-scintillation gain, the extraction yield from the liquid to the gas phase and the electron drift velocity.