It all depends on the rules by which each system works.

In Thunderbird you mark it as Not Junk to train Thunderbird that it is in fact a good message. There ought to be a button in the message to set it as Not Junk, or you can select it in the message list and use shift+j to mark it as Not Junk. Right-click will also offer options for marking the message.

You'll have to check with aol as to how they do it.

In Gmail, yes moving it back to the Inbox should set it as Not Spam. That's nice because you don't have to go to their website to communicate your wishes to them.

In yahoo, it's pretty much all futile; if they say it's spam there's nothing you can do about it.

Some mail services may allow you to whitelist senders or use a checkbox or similar to mark it as not spam. You have to "read the fine manual". ;-)

Your account settings in Thunderbird have adjustments for how long to retain junk messages. The only reason not to delete them immediately is to give you a chance to review the Junk folder and rescue any messages that shouldn't be there. Once Thunderbird has analysed the message and counted the tokens in it, you don't need to keep the message. Mine are set to delete after 7 days since it's unlikely that I will have not already saved the good messages in that time.

As to how Thunderbird does it...

It counts word occurences in messages. It logs the words that appear most often and correlates them with your scoring as Junk or Not Junk. Gradually it will associate junk messages with the words found in junk messages, and good messages with good words. It pays no attention whatsoever to the sender, and scarcely any to the subject line. Senders are not trustworthy markers of spam, since no sensible spammer would send from his own computer, but would send from "owned" computers using stolen "from:" addresses. Hence the from: address has effectively zero value in recognizing spam.

The "database" in Thunderbird is a file named training.dat. If you want to learn more about this, look for the Junquilla addon.

I can't get any incoming email to go to my inbox. I believe I have checked all the appropriate boxes. Account settings, Junk settings, and checked the box for "Do not automatically mark mail as junk if sender is in: and I checked the box for "Personal Address Book". I even used the upper case 'J" to indicate that it is not junk. At a loss for what to do next. Thanks for your help.