Alex runs a big public cookbook on GitHub. It has waffles, crepes, and french toast. No pancakes. Somebody has to fix this.
You can't push to Alex's project, because it isn't yours. A fork is your own copy of it on GitHub, and you can push to that one all you want.
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A fork lives on GitHub, under your name. It remembers where it came from, which is how a pull request finds its way back to the original later.