Hibernation is an inactive computer state in which ram contents have been saved to non-volatile memory (such as a hard-drive). This allows the computer to completely power down, while maintaining running programs.
Hibernation serves the same purpose as sleep mode. Hibernation takes longer to suspend and resume, but uses no power (so a hibernated computer can be stored indefinitely or its battery can be removed).
Windows PCs and Macs have a feature ("Hybrid Sleep" on Windows) which allows a combination of sleep and hibernate where the computer sleeps but also copies the contents of the RAM to disk, in case of power failure.