Some web sites can prevent browsers from saving password information. Некоторые веб-сайты можно запретить сохранение паролей браузеров.

English; Go to the Mozilla Add-ons Web Page https://addons.mozilla.org/en-US/firefox/

русский; Перейти к Дополнений Mozilla веб-страница https://addons.mozilla.org/ru/firefox/

(There’s a lot of good stuff here) and search for Password. (Есть много хорошего материала здесь) и поиск Пароль.

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.
• Options > Privacy > "Use custom settings for history" > Cookies: Exceptions

In case you are using "Clear history when Firefox closes":

• do not clear the Cookies
• do not clear the Site Preferences

• Options > Privacy > Firefox will: "Use custom settings for history": [X] "Clear history when Firefox closes" > Settings
• https://support.mozilla.org/kb/remove-browsing-search-and-download-history

Note that clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, passwords, and other website specific data.

Clearing cookies will remove all specified (selected) cookies including cookies with an allow exception that you would like to keep.