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I am not a trader myself but am trying to educate myself about trading.

I notice that in most articles and videos about technical trading, an illustrative graph is displayed showing the latest price of a stock. This price (call it P) I presume corresponds to the latest price agreed between a buyer and a seller. Also presumably, there are a collection of currently-unmatched buyers and sellers that are hoping to buy/sell at prices slightly above or below P. They may declare "I have X amount of shares to sell if anyone will buy them for \$Q each" or "I will buy Y shares if anyone will sell them for \$R each".

I don't know what these declarations are called, are they "orders"? I presume that many orders are never matched, or are only partially matched or may get cancelled.

My main question is: can everybody see the current set of unmatched orders? Or is it only the people that actually run the exchange? And if it is visible, is it the kind of data that is readily available at reasonable cost or the kind of thing that only the big banks and HFT traders would ever get access to?

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Unmatched buyers and sellers prices are called bid/ask offers. What you are referring to is the order book. Yes, this is something you can see in real time if you subscribe to it with your broker (L2 quotes, should be well under 100$ a month). L1 quotes are the 1st level bid/ask and the recorded transactions while L2 quotes are for the full order book including 2nd+ level of bid/ask.

That being said, the order book is being filled with bogus information since the last couple of years so it's pretty hard to read and only useful to a certain degree. The big banks and HFT traders have access to the same information but since their servers are sitting right next to the exchanges, it available to them (and their algos) faster by a few milliseconds.

Some reading I recommend (very entertaining book even for the non-traders) https://www.amazon.com/Flash-Boys-Wall-Street-Revolt/dp/0393351599

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