City of Bones
You've seen this kind of psychic before. You know, the kind with the sign that says "Pyshic Readings" [sic] or something similar, and you think "She must have known the sign would be misspelled, haha." Madame Dorothea seems like that kind of fortuneteller at first glance, but she's a lot more magical than you've given her credit for.
As we said, at first Madame Dorothea seems a little batty, always wanting Luke to change her light bulbs and stuff. But when Clary visits her after being attacked by the Forsaken, she realizes that not only does Madame Dorothea know Jace, who calls her "Lady of the Haven" (18.238), she also knows about the Forsaken, and Clary's mother's history. Plus she's a great source of cryptic foreshadowing, predicting Jace and Clary's forbidden love, among other things.
Her Tarot deck ends up being a critical element in the plot, as the Ace of Cups in the deck, painted by Clary's mother, actually holds the Mortal Cup within it. When Clary retrieves it, Madame Dorothea is possessed by the demon Abbadon, which kills her. Bet she didn't see that coming. (Sorry, we couldn't resist.)