OK, but even with a private key, one must know where to use it, right? I cannot just take that private key, plug it into my poloniex/bittrex/whatever wallet and receive the funds, right? Or does it work that way?
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It works exactly that way.
Also if you think that's scary, consider that your private key is actually just a number between 0 and 2^256-1 which means anyone that guesses it also has access to your funds.
Fortunately that is still something on the order of a trillion keys for every atom in the known universe. So it's a really, really large space to search...