We consider the phenomenon of afterpulsing in avalanche photodiodes (APDs) operating in gated and free-running Geiger mode. An operational model of afterpulsing and other noise characteristics of APDs predicts the noise behavior observed in the free-running mode. We also use gated-mode data to investigate possible sources of afterpulsing in these devices. For 30-um-diam, 1.06-um-wavelength InGaAsP/InP APDs operated at 290 K and 4 V overbias, we obtained a dominant trap lifetime of td=0.32 us, a trap energy of 0.11 eV, and a baseline dark count rate 245 kHz.