^ When in the Shutdown screen press the Shift key, the Standby button will be replaced by the Hibernate button.
The standby option stores the computer "context" in RAM, so you need to keep the computer on. With Hibernate the computer "context" will be stored in the hard drive so you don't need to keep the computer on.
Regarding your firefox question just terminate the firefox with task manager, not closing it, and the next time you run firefox it will detect that the firefox hasn't been correctly closed and it will ask you if you want to restore the previous session.
It's a lamme answer but it works