Hibernation is if you have to shut the machine off, but don't want to wait for the full cold boot load time/are in the middle of something important and want to save your progress even though you have to shut down.
What it does is dumps a copy of what's actually in
ram into the hibernation file on your disk along with some extra state information so that when you next boot, it's just like returning to what you were doing. Nothing 'continues' in the background or anything like that because everything is frozen at that point.
I use this if I'm working on something important on my laptop at home that won't really 'save' well, and need to take the lappy with me on a LONG drive before I'll have a chance to work on it again (basically, if it's something I don't wanna lose my place on when I go to or from work) I'll hibernate instead of shutting down.
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