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When you buy a stock, you don't get a physical stock certificate. You broker gets confirmation of your "ownership" and maintains it for you.

This I did not know.

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Those same brokers maintain Stock Loan Departments which are nothing more than stocks available for loan at a price.

At what price? Does the broker set the price? Because as I understand, the hedge funds have an estimate what the stock price will eventually be so why is the broker setting the price? Or does the borrower wait until the broker offers a "loan" of the stock agreeable to the buyer's estimate of what the stock will eventually be?

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If I want to short a stock, I contact a stock loan department to borrow it. If I approve of the financial terms, I borrow it and sell it hoping that, when I have to return it to the Stock Loan Department, I'm buying it back at a lower price.

Does the broker tell you when you must return the stock? The buyer does not decide this? And the borrower immediately sells the stock and awaits a time to pay for it? When the time is right, however decided, the "loan" is the stock price at the time of payback? If I paid $100 a share when I borrowed the stock but it is only worth $50 a share when I pay back, then I have made $50 a share?

I am trying so hard to understand this. And I do invest in the stock market but through a mutual fund type thing.

Shorting a stock has a time limit. You owe that stock back later, plus a fee.

If you borrow the stock at $20, thinking it will go down, and it does go down to $10, you buy one share at $10 and keep the $10, minus perhaps a $1 fee, for a $9 profit.

If the stock goes up, you lose money.

This is more about naked shorts, which are illegal. It means someone pretends to give you stock to short but they don’t actually have any. It is a fraud contract on the market. This is what is happening here. Hedge funds have fraud contracts they’sold’ to each other; illegal naked shorts. Naked, because the contracts have no real stock behind that that is involved.

When the broker loans the stock, the fee being charged is like the interest on a loan. What that interest rate is will be determined by your credit worthiness and supply-demand interest in the stock. The longer you (the borrower) keep the loaned stock the higher the cost to you as the interest builds up. So the "price" I referred to is not the target price of the stock but the fee you're paying for borrowing.

Normally, the broker wouldn't tell you it has to be returned by a fixed date, but as in the case of GME, the fees you pay on the loan may be variable so it becomes a game of how long can you hold your breath underwater. To use your example, if you borrowed a stock at $100 to sell, repurchased at $50 to close out the short your profit would be $50 less the fees you paid for the loan. Those fees could easily wipe out your "profit."

Typically, you the private investor do not “buy” the stock.

You contract the broker to act as your agent, who buys the stock.

They buy the stock with your money, and have some rights of ownership. For instance, there are times they can liquidate “your” stocks to cover “their” bets in order to protect the system.

Unless you have something physical in your hand, do you really own it?