Miss Peregrine is, unsurprisingly, the headmistress of Miss Peregrine's Home for Peculiar Children. Who do you think would be in charge there? Miss Hannigan? Because of Jacob's dying grandfather's clues, Jacob thinks he has to find Miss Peregrine, who must be almost one hundred by this point, in order to get answers about Grandpa's life.
Jacob's partially right: He does have to find Miss Peregrine, but she's still in her late thirties/early forties, thanks to the time loop. Plus, she can turn into a bird. A peregrine, of course, which is "a kind of hawk" (3.110). She uses her birdly powers to spy on Jacob in bird form (it's either that or shapeshift into Edward Cullen) and she can manipulate time.
She's a nice lady, taking in peculiar children (she once was one herself, after all) because the world can be dangerous for them. But we don't get to learn too much about her background. She has two evil brothers who are the reason why the monsters, known as the hollowgast, torment the peculiars, so maybe there will be a dramatic family showdown at some point. But in this book, Miss Peregrine is forced into bird form, along with Esmerelda Avocet, her mentor, and both are kidnapped. Miss Peregrine is recovered, but still remains birded at the end of the book.