Using NASA's Submillimeter Wave Astronomy Satellite (SWAS), we have examined the production of water in quiescent and shocked molecular gas through a survey of the 556.936 GHz 110-101 transition of ortho-H2O in the NGC 1333 molecular core. These observations reveal broad emission lines associated with the IRAS 2, IRAS 4, IRAS 7, and HH 7-11 outflows. Toward three positions we detect narrow (Δv ~ 2-3 km s-1) emission lines clearly associated with the ambient gas. The SWAS observations, with a resolution of ~4', are supplemented with observations from the Infrared Space Observatory (ISO) and by an unbiased survey of a ~17' × 15' area, with ~50'' resolution, in the low-J transitions of CO, 13CO, C18O, N2H+, CH3OH, and SiO. Using these combined data sets, with consistent assumptions, we find beam-averaged ortho-H2O abundances of greater than 10-6 relative to H2 for all four outflows. A comparison of SWAS and ISO water data is consistent with nondissociative shock models, provided the majority of the ortho-H2O (110-101) emission arises from cool postshock material with enhanced abundances. In the ambient gas the ortho-H2O abundance is found to lie between 0.1 × 10-7 and 1 × 10-7 relative to H2 and is enhanced when compared to cold prestellar molecular cores. A comparison of the water emission with tracers of dense condensations and shock chemistry finds no clear correlation. However, the water emission appears to be associated with the presence of luminous external heating sources that power the reflection nebula and the photodissociation region (PDR). Simple PDR models are capable of reproducing the water and high-J 13CO emission, suggesting that a PDR may account for the excitation of water in low-density undepleted gas, as suggested by Spaans & van Dishoeck.