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:

• 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

You can remove the cookies from a website to make the website forget about you.

If you want Firefox to store the password, the easyest way is to log off, then back on. And let Firefox record the password.