This page is a work in progress. If you're looking for any additional information about using the hibernation feature in Windows, just scroll down to the bottom and add a comment. I'll do my best to answer all questions and update this resource.
What Is Hibernation?
Hibernation gives you the ability to take a "snapshot" of the current state of your computer before shutting down. The next time you start up your computer, everything will (or should) be exactly the same as it was before choosing to hibernate. It's very similar to "standby" mode, which puts your PC into a low power mode. Hibernation goes one step further than standby though by writing the contents of the RAM onto the harddrive and then competely powering down everything.
It's good for the environment (and your electrical bill) – if you're not going to be using your computer, there's not much reason to have it still eating your electricity.
I use it to speed up the startup time of my laptop. A fresh power-on can take anywhere from a minute or more, but starting up from hibernation will give me a usable system in about 20 seconds.
The option to enable hibernation is located in within the "Power Options." Power Options can be found within the Control Panel. Hibernation gets its own tab within the Power Options containing only one option – a checkbox labeled "Enable hibernation." Most, if not all, modern hardware should support hibernation, so you might as well give it a try.
Also inside the Power Options is a tab to configure your Power Schemes. These settings allow you to set the time your PC is inactive before turning off the monitor, spinning down your harddrives, going into standby mode, and even hibernating. The ideal settings for these are going to vary based on your hardware, usage, personal preferences, etc. If you've never taken a look at this and you're using a laptop, you might be able to stretch your battery life a little longer by adjusting some of these.
How To Hibernate
Based on the searches leading people to my hibernation post, it would appear that quite a few people are unable to find the hibernate option. There are several possible reasons for this, the most likely being that hibernate wasn't enabled.
When hibernation is enabled, it is listed under the Shutdown menu. Depending on how your system is configured, you may be presented with two different screens when you choose to shutdown. One of these screens lists the shutdown options in a dropdown menu. Just choose hibernate and you're on your way. The other screen shows the shutdown options next to huge icons. I've never seen hibernate listed on this menu so it may or may not be there. If it doesn't show up, pressing the "h" key on your keyboard will select it.
Problems With Hibernation
Sometimes I'll run across something that doesn't quite work right after starting up from hibernation. It could be a program that doesn't want to play nice, or hardware that just doesn't seem to get it. This is rare, and seems completely random – usually it can be fixed by using the Task Manager to terminate the offending program, or unplugging and replugging in a bluetooth dongle, etc. In my experience, the 'worst' problems can always be resolved by a clean restart of the system.