Windows PCs typically operate in a number of Advanced Configuration and Power Interface (ACPI) power states. An S0 power state, for example, is when your PC is running and ready to respond to your input. There are a number of sleep states, including S1, S2, and S3, and there's also a hybrid sleep state where hibernation is used in tandem with a sleep state.
Hibernation is considered an S4 power state. While hibernating, your PC will seem like it's completely off, but there will be a saved hibernation file ready to be used to boot back to where you were during your last user session. Some power is usually still routed to peripherals so that you can, say, tap your keyboard and have the PC boot.
An S5 power state (soft off) is when your PC is shut down and rebooted completely. There's no hibernation file and no saved user session. There is also a G3 power state, which is when your PC consumes absolutely no power and is by all means off.
With fast startup enabled, choosing to shut down your PC might look like you're completely shutting things down, but in reality, your PC is entering a mix between a shutdown and hibernation. A hibernation file is indeed used, although it is smaller than usual. Why? You're logged off before the file is created, meaning your session is not recorded. The speed boost comes from the Windows kernel being saved on your hard drive and loaded when booting.