The photodiode has a depletion region at zero bias, but it becomes thicker with applied reverse bias. With reverse bias, there is more volume to stop photons (more important at higher energy than visible), the capacitance decreases, but the leakage current increases (especially at high temperature). At zero bias, the device looks like a resistance and capacitance in parallel with the photocurrent generator (the resistance is the slope of the I-V curve through the origin), and the resistance is a noise generator. However, the dark current (and its noise) goes away.
In reverse bias, with a high-impedance load, the capacitance charges up until the voltage reaches the bias and the device conducts further current away. This is common in arrays, but the capacitance is non-linear. If you discharge the device into a charge-sensitive amplifier, the output is more linear than if you only read the voltage.
The device can be run into a virtual ground with zero or reverse bias to measure photocurrent.