Saved passwords are stored under Firefox Preferences > Security > Saved Passwords. There you will find a list of saved passwords and what website they were saved for. You can view the passwords or delete them at will.

Details like websites remembering you (log you in automatically) are stored in a cookie.

• Create a cookie 'allow' exception to keep such cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Tools > Options > Privacy > "Use custom settings for history" > Cookies: Exceptions

If clearing cookies didn't help then it is possible that the cookies.sqlite file that stores the cookies got corrupted.

• Rename (or delete) cookies.sqlite (cookies.sqlite.old) and delete other present cookies files like cookies.sqlite-journal in the Firefox profile folder in case the file cookies.sqlite got corrupted.

• http://kb.mozillazine.org/Cookies
• https://support.mozilla.org/kb/Deleting+cookies

You can remove all data stored in Firefox from a specific domain via "Forget About This Site" in the right-click context menu of an history entry ("History > Show All History" or "View > Sidebar > History") or via the about:permissions page.

Using "Forget About This Site" will remove all data stored in Firefox from that domain like bookmarks, cookies, passwords, cache, history, and exceptions, so be cautious and if you have a password or other data from that domain that you do not want to lose then make sure to backup this data or make a note.

You can't recover from this 'forget' unless you have a backup of the involved files.

It doesn't have any lasting effect, so if you revisit such a 'forgotten' website then data from that website will be saved once again.