We investigate electrical transport and optoelectronic properties of field effect transistors (FETs) made from few-layer black phosphorus (BP) crystals down to a few nanometers. In particular, we explore the anisotropic nature and photocurrent generation mechanisms in BP FETs through spatial-, polarization-, gate-, and bias-dependent photocurrent measurements. Our results reveal that the photocurrent signals at BP-electrode junctions are mainly attributed to the photovoltaic effect in the off-state and photothermoelectric effect in the on-state, and their anisotropic feature primarily results from the directional-dependent absorption of BP crystals.