Hibernation, or suspend to disk, means powering down a computer while retaining it's state. The memory contents are typically saved to a SSD or hard drive upon hibernation. That content can be read back to restore the machine to its previous state when it is powered back on.
Advantages And Disadvantages Compared To Suspending To RAM
Linux and other operating systems support suspend to RAM. This allows a machine to power down most components such as SSDs, HDDs, graphics cards while retaining the machines state a minimum amount of power to keep the RAM contents alive.
Suspending to RAM has one clear advantage over hibernation: It's a lot faster.
There's also one not so obvious huge disadvantage: Encryption keys and other contents remain available in RAM while the machine is turned off. This makes it possible to do cold boot attacks. A machine that's suspended to RAM is as exposed as a powered-on machine in that regard.
Linux lets you combine suspend-to-RAM with disk hibernation. This can be specially useful for laptops. This mode saves the RAM contents to disk while keeping the RAM content available in RAM when the machine is powered off. The machine in hybrid suspend can be quickly resumed unless the battery runs out (or power is cut in case of a desktop). Resuming to the same state as the machine was in before hybrid suspend will still work even if the machine has lost power while it was hibernated.
The ACPI specification
The ACPI specification defines suspend to disk as mode S4.
Quirks And Issues
Suspend to RAM and disk is generally supported on Linux. However, there are some combinations of motherboards, processors and graphics cards that have issues with either suspend to RAM or suspend to disk or both.