Some sites, especially banking, medical, insurance, and shopping sites, will log you out after a certain period of inactivity..... Others, that aren't involved with your money, or health, may let you stay logged in indefinitely.
If I forget to log out after a session, Amazon usually greets me by my username, but makes me enter my password to view or make transactions. If I remember to log out, it needs both username and password, before I can do anything serious.
If my IP address has changed, my bank also asks one of the "secret questions" I signed up with.
Check to make sure you've "allowed" Firefox to save cookies for the sites you want to remain logged into:
TOOLS - Options - Privacy & security - Cookies and Site Data - Manage exceptions
