The Too-Personal Story
This is the one that reveals too much about a writer in some way that could be potentially embarrassing. Perhaps it's just an anecdote that you found funny and you wanted to share it because - well, because when you laugh at something you feel better. Even if it's nervous laughter. Everyone has biases they know are inappropriate and the vast majority of us work hard to overcome those biases. What writers have to be careful of is revealing those biases too obviously in their work. Writers can be a terribly self-aware lot, superstitious and cowardly, too, and the common defense to a bias revealed in the work is the writer saying "It's only a character, not me and not how I really feel". And sometimes that's true. I leave it as an exercise for the reader to determine when it isn't.
The You Had To Be There Story
Sometimes a writer is able to recreate the sequence of events that actually happened leading up to what amounts to a centerpiece. There are heart-wrenching tales that make readers cringe in sympathy (you've got examples you're probably thinking about right now) and there are silly circumstances that make readers laugh and sometimes those stories are true but the names and locations have been changed to protect the writer from a lawsuit. Every writer knows that even True Stories told for public consumption have to be amped up, the angles of the drama skewed just a little in order to make it more compelling. This may be the writer's attempt at retelling a college anecdote in a new way that disguises what really happened. However, if you weren't there, you can't fully appreciate what it is that made the writer want to tell it and I imagine that if you didn't share that experience with the author you wouldn't find the original version at all compelling.
You know how when someone starts telling a story in a party situation and they stutter and falter and start saying things like "Wait, I'm not getting this right"? Yeah. That's the story that shouldn't be told by that person. Ever.
The Too-Specific Story
This, I suppose, might be considered an alternate version of The You Had To Be There Story. This is a story that has very specific references that will not make any sense to the reader. Usually it's in the setting or in a poor description of the characters involved. The reader is left scratching his head thinking that he missed something. Sometimes he'll go back and re-read something in order to ascertain if that's the case or if the writer just didn't connect somehow. Most often that story is dismissed and never thought of again.
The Story You're Dying to Tell
And this is the point: you may not be ready to tell the story that's been burning in your head for a long time. Patience, grasshopper. Learn the craft, make the mistakes, get some experience before you try to tell the whopping great yarn you've got stuck in your head. Spend time developing it. I've got a series that I've been noodling with over the last fifteen years that I'm still not ready to tell but it's there on paper in my files, some copies are digitized on my computer and every time I open them up, I add something I've learned since the last time, some detail or nugget. I've tried to tell this story as a comic book and as a radio play but those weren't the right formats for it. This is a series of novels and I really want to get it out. So I keep notes until I'm ready to start typing.
And now that I've built it up and discussed it in public I'll have to release it some day. Not yet, though. I'm not ready to tell it. I don't have the right experiences and perspective. I will. And maybe sooner than I think. Or maybe not.