Let me take a shot at explaining continuations.

In normal programming, functions "return" their values. In Continuation Passing Style (CPS), functions never return. Instead, they take another function as input; and instead of returning, they call that function (the "continuation"). Instead of returning their output, they pass their output as input to the continuation.

(Some optimizations are used such that this style of call, the "tail call", does not cause the stack to grow endlessly.)

Why would you write code in this style? Generally, you wouldn't. It's typically used as an internal transformation in some types of interpreters or compilers. But conceptualizing control flow in this way has certain advantages.

Then there are terms like the "continuation" of a program at a certain point in the code, which just means "whatever the program is going to do next, after it returns (or would return) from the code that it's about to execute". That's what "call with current continuation" (call/cc) is about. It captures (or reifies) "what will the program do next after this?" as a function that can be called to do, well, do that thing. If your code is about to call `f();`, then the 'continuation' at that point is whatever the code will do next after `f()` returns with its return value.

Thus if you had some code `g(f())`, then the continuation just as you call `f()` is to call `g()`. CPS restructures this so that `f()` takes the "thing to do next" as input, which is `g()` in this case. The CPS transformation of this code would be `f(g)`, where `g` is the continuation that `f` will invoke when it's done. Instead of returning a value, `f` invokes `g` passing that value as input.

You can use continuations to implement concepts like coroutines. With continuations, functions never need to "return". It's possible to create structures like two functions where the control flow directly jumps between between them, back and forth (almost like "goto", but much more structured than that). Neither one is "calling" the other, per se, because neither one is returning. The control flow jumps directly between them as appropriate, when one function invokes a continuation that resumes the other. The functions are peers, where both can essentially call into the other's code using continuations.

That's probably a little muddy as a first exposure to continuations, but I'm curious what you think. I generally think of continuations as a niche thing that will likely only be used by language or library implementors. Most languages don't support them.

Also, I'd probably argue that regular asynchronous code is a better way to structure similar program logic in modern programming languages. Or at least, it's likely just as good in most ways that matter, and may be easier to reason about than code that uses continuations.

For example, one use-case for coroutines is a reader paired with a writer. It can be elegant because the reader can wait until it has input, and then invoke the continuation for the writer to do something with it (in a direct, blocking fashion, with no "context switch"). But you can model this with asynchronous tasks pretty easily and clearly too. It might have a little more overhead, to handle context switching between the asynchronous tasks, but unlikely enough to matter.