Lol, you pretty much answered your own question with what you wrote on the bottom.
Yes a fanfic, is a story/fictional tale written by a fan, it can be about ANYTHING at all, and is only limited to the imagination of the writer and what he can imagine and write.
It doesn't at all have to be about true events, but can be about whatever you want it to be. Action/adventure/drama/romance/horror/etc.
And of course it mainly involves Hello Project girls as the characters in the story, sometimes told from a 3rd person perspective, or from a first person perspective, where the writer tells the story from a specific girls point of view and what happens to her.
They're really fun to write and completely fictional, well most of it anyway, sometimes daily events in H!P or recent news is thrown into the story, or recent news can help create a new story all together, like what the writer imagines happens to the H!P girls from their point of view.