School: Headmistress of Beauxbatons
Madame Maxine is the headmistress of Beauxbatons, a wizarding school that appears to be located in France (though it's Unplottable, so the actual location is secret. But everyone from there speaks French, so we're going with France). She's also enormously, enormously tall. The first time Harry sees her, he thinks that the only person that he has ever seen who is as tall as Madame Maxine is Hagrid, the Hogwarts gamekeeper. In spite of her enormous height, Madame Maxine is handsome and graceful, and, as Professor Dumbledore notes, "an excellent dancer" (29.155).
As soon as Hagrid sets eyes on her, he seems to be smitten. He starts dressing in some truly horrible suits to try and impress her. Hagrid also brings her (and Harry, in his Invisibility Cloak) to see the dragons so that she can tell Fleur what the first Triwizard task will be. And during the night of the Yule Ball, Hagrid takes Madame Maxine aside and tells her that he's like her: half-giant. Madame Maxine doesn't respond to Hagrid's confession at all well, though. She tells him, "'Alf-giant? Moi? I 'ave – I 'ave big bones!" (23.204).
Madame Maxine's refusal to admit the obvious – that she's also half-giant – puts a real strain on the budding relationship between herself and Hagrid. Hagrid starts stomping around making fairly obvious remarks about the importance of never being ashamed: "Big bones ... I'll give her big bones" (24.192). But, by the end of the novel, Madame Maxine seems to have come around. She and Hagrid make up, and Hagrid hopes that he can convince Madame Maxine to go with him on a mysterious summer mission for Professor Dumbledore. What could it be?
Aside from her role as Fleur Delacour's supporter, Madame Maxine is important in the novel because she gives Hagrid a reason to talk about his half-giant heritage in public. Without Madame Maxine to listen to his confession, how would Rita Skeeter have discovered Hagrid's secret? So, Maxine's a useful tool to get some information out there about Hagrid's past that's going to prove important in future novels – especially in Order of the Phoenix.