Whenever the Pilot appears to be turned off, it is actually sleeping, but still running. The system puts as many peripherals and other parts of the hardware to sleep as possible, to minimize power consumption, and then monitors the hardware keys and the real-time clock for any activity. If a key such as the on-off key is pressed, the CPU wakes everything back up again, ready to run. A Pilot can sit for months in sleep mode. Sleep mode isn't remarkable.
Most handheld computers have an equivalent operating mode.
Doze mode, on the other hand, is quite clever. The Pilot processor actually stops running, but peripherals like the screen stay awake, so that the system can respond quickly, in a fraction of a second, to an event like a pen tap. Whenever the Pilot appears to be running, but nothing is happening, it's probably in doze mode. The Pilot actually dozes in between your pen actions, even when you are entering characters on the Graffiti pad or using the on-screen keyboard. It just looks asleep, and wakes up very fast, so fast, you don't even know it's sleeping.