I tried to put in that address but it doesn't save it.

It appears to save it, but when I close the browser and open it again it isn't there.

That implies that the cookies are removed when the Firefox "session" ends which explains their "at end of session" status. This means that they are no longer available after restarting the browser.

Two options: Either set that to "until they expire" (which would retain any other website's cookies as well beyond the session) or click on "Exceptions" and create a respective entry for forums.mozillazine.org to allow cookies to persist beyond the session.

Why, yes, I have it set to 'keep until I close Firefox'.

In the Privacy options, do you have "Accept cookies from sites" set to "Keep until: [I close Firefox]" by any chance, or defined an Exception to allow session cookies for mozillazine.org only?

Using an icon I found on my bookmarks bar that says 'Opens the cookie manager for the current web page', I found four cookies and deleted them. Restarted the browser and it still didn't log me in automatically.

All the cookies I deleted said 'Expires: at end of session'.

First, try removing any mozillazine cookies and re-accept them.

Moving to Site discussion.

When I log in to this site (this site, Mozillazine) there is a box to check that says

'Log me on automatically each visit'

I check it and log in, and yet on subsequent visits it will never log me in automatically.

What am I doing wrong?