Are you allowing these sites to set persistent cookies? Many sites save passwords in cookies.

Make sure that you do not run Firefox in (permanent) Private Browsing mode.

• https://support.mozilla.org/kb/Private+Browsing

• Tools > Options > Privacy: Use custom settings for history
• Deselect: [ ] "Always use private browsing mode"

Do you mean names and passwords in the Password Manager or do you mean that you are no longer logged on to (remembered by) websites after closing and restarting Firefox?

There is a difference between remembering the name and password in the Password Manager and a "remember me" check box on a web page to log you in automatically.
The latter usually involves the creation of a special "remember me" cookie that is stored on your computer and that is send to the server.
If that "remember me" cookie is removed or gets corrupted or is not send to the server then the server won't remember you anymore and you need to sign in once again.

• http://kb.mozillazine.org/Cookies
• http://kb.mozillazine.org/Password_Manager

Websites may be using autocomplete=off to prevent Firefox from saving and filling the name and password.
You can remove autocomplete=off with a bookmarklet to make Firefox save the name and password.

• http://kb.mozillazine.org/User_name_and_password_not_remembered

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

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

• Tools > Options > Privacy: History: [X] Clear history when Firefox closes > Settings
• https://support.mozilla.org/kb/Clear+Recent+History

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

I don't use Password Manager, I'm no longer logged in on sites. When I go to the privacy section, I am not getting the options you have shown.

Then you are not using the custom settings in Privacy in the Options.

See also:

• https://support.mozilla.org/kb/Options+window+-+Privacy+panel