Is the password saved, but disappears? Does the browser ask if you want to save the password? Are you using any password add-ons/programs?

Profile data can be removed or not recorded several ways;

• 1) Running Firefox in Private Mode (nothing is recorded)
• 2) Privacy settings; (In the address bar, type about:preferences#privacy<enter>)
• 3) Many disk cleaners and anti-virus type programs can remove data from browsers. Open their settings and have them leave the browsers alone.
• 4) Mal-ware.

Type about:preferences#security<enter> in the address bar. Look under Logins,

Note: For Firefox 56+ its; about:preferences#privacy Look under Forms & Passwords +++++++++++++++++++++++++++++++ Turn on Remember Logins. To check the passwords you have listed, press Saved Logins.

Press the Exceptions button and see what's inside.

A detail like a website remembering you (log you in automatically) is stored in a cookie.

You can use these steps to make a website recognize and remember you:

• you can create an "Allow" exception to keep specific cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Options/Preferences -> Privacy & Security
  Cookies and Site Data: Exceptions

Removing cookies from websites makes the website forget about you.

In case you use "Clear history when Firefox closes" or otherwise clear history.

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

• Options/Preferences -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an "Allow" exception you may want to keep