Firefox includes a Password Manager that can save the passwords you use to log in to websites. This article describes why your passwords might not be saved.
Firefox will remember passwords by default. You may have disabled this feature, or told Firefox to never remember passwords for a particular site.
Now that you've configured Firefox to remember passwords, try logging in to the site again.
If you use the Firefox Private Browsing feature, no passwords will be automatically filled in for your Private Browsing window or in permanent Private Browsing mode, and no new passwords will be saved. Check you are not in a Private Browsing window:
You may also be in permanent Private Browsing:
Firefox may be set to remove saved passwords when it is closed. To undo this setting:
Firefox is now configured to not clear your passwords when closing.
You may have Password Manager software, maybe as a feature included in security software, that prevents Firefox from storing passwords. Check your Password Manager settings. For more information see Where are my logins stored?
Some graphics driver versions may have problems to display prompts. To see if this applies to you, follow the steps in Troubleshoot extensions, themes and hardware acceleration issues to solve common Firefox problems.
Some websites do not allow for passwords to be saved for security reasons. If you have followed the steps above, but are still not prompted to save a password when logging in to a certain website, that site may have disabled password saving.
Based on information from User name and password not remembered (mozillaZine KB)