The easiest way to think about continuations is to consider them a generalization of function returns. The continuation of a C function f() is the return address
and the saved frame pointer of the calling function -- and that looks a lot like a closure, and that's because it is, except that a) you can only pass that closure one argument in C: the return value, and b) you actually can't get a value for this closure in C, and c) there's no closures as such in C anyways :)
[And what is a closure? It's a tuple of {function body pointer, data pointer}, and "callbacks" in C often look just like that, but not as a singular value combining the two parts but as two separate values. In better languages a function is indistinguishable from a closure.]
Anywhere that you see `call/cc` you can think of it as making a first-class function value (a closure) of the current location (which would be the return address of the call to `call/cc`) and then passing that closure to the function given as an argument to `call/cc`. After you parse that sentence you'll realize that's a bit crazy: how does one make a function out of... a function's return address (and saved frame pointer)? The answer is: reify that {return address, saved frame pointer} as a closure. ('Reify' means, roughly, to make a hidden thing not hidden.)
Still, it must seem crazy. Yet it's not that crazy, certainly not if you make it so you can only call it once, and only if `call/cc` hasn't returned yet. What's crazy is that in Scheme this continuation can be called repeatedly, so how? Here's one way to do it: replace the stack with allocating function call frames on the heap!, which in a language with a GC means that all those function call frames remain alive as long as closures (which a continuation is) refer to them. (Another way to do it is with copying stacks.)
One can easily (for some value of "easily") implement a language that has `call/cc` in a language that doesn't but which has a) closures, b) a powerful macro system / AST / homoiconicity. All one has to do is take a program, apply continuation passing style (CPS) conversion to it (a fairly straightforward transformation where the result is unreadable for most humans), which automatically causes all function call frames to be captured in closures (thus put on the heap), but also automatically makes continuations explicit values (closures). The `call/cc` is a trivial function that passes its now-explicit continuation argument to the function that `call/cc` is given to call.
Allocating function call frames on the heap is a performance disaster. And that's where delimited continuations come in: the goal being to allocate function call frames on mini stacks.
`call/cc` is really a bit of a parlor trick, and a very nifty one at that, but continuations are a very important concept that comes up all over the place, and one that computer scientists and software engineers ought to be at least conversant with.