What does shutter latency mean on a Phase One camera, and when should I use each setting?

I don't know medium format cameras, but in general shutter latency is the time that it takes from when you push the shutter to the time the shutter actually activates. The lower the latency, the more responsive the shutter, but the less time the camera has to make last minute adjustments or changes in hardware state. A higher shutter latency will allow the camera to make finer tuned adjustments and turn on hardware that was sleeping, but will not go as close to when you push the shutter.

Some other cameras express this as shutter priority vs focus priority. Lower latency should be used when you need the shot at the exact moment you push the shutter where as longer latency will generally let the camera do more work for you or leave stuff powered down and may generate a better quality image or increase battery life.

It may also do things like mirror lockup to reduce the latency so that the shutter can fire faster or keep hardware turned on.