Say I write a story, or even a book in chapters on a forum, in the form of a Choose Your Own Adventure, is my story or book protected by copyright there or can anyone copy paste it and publish it under their own names?
Depending on where you live, you own the copyright to your work the minute you write it.
Simply posting your work online does not give readers the right to copy it. Alas, it does give people the ability to copy it. But not the legal right.
Carefully check the details of the forum where you post the work. Your user agreement will almost certainly give the forum certain rights. At very least, it will give them the right to display your work to other forum users. After all, that's necessary to the function of a forum.
Usually online forums need only the right to store your work and display it to forum readers. But read the user agreement very carefully, double-checking what rights they require.