Madame Maxime is the extremely tall headmistress from the Beauxbatons Wizarding School.
Hagrid takes a shine to her and her statuesque figure, and the two seem to have quite a flirtation going throughout the film. We see that in their gooey dialogue when Hagrid calls her out into the woods late at night to show her something:
HAGRID: Bonsoir, Olympe.
MADAME MAXIME: Oh, Hagrid. I thought perhaps you weren't coming. I thought perhaps you had forgotten me.
HAGRID: Couldn't forget you, Olympe.
MADAME MAXIME: What is it you wanted to show me? When we spoke earlier, you sounded so…exhilarated.
Okay, get your minds out of the gutter: Hagrid was just showing her some dragons. But from their back and forth, you can tell that they totally heart each other.
Beyond the flirtation with Hagrid, we don't get to know too much about Madame Maxime—but if Hagrid likes her, that's good enough for us.