"When there's something strange, and it don't look good—who you gonna call?" Ghostbusters? Well, yes. But a flying nanny works just as well.
When the movie opens, The Banks Family is in distress. The family nanny has just quit, and the two kids, Jane and Michael, have run away from home (though they actually just got lost chasing a kite). Their uptight dad, George Banks, decides to put out an ad for a strict no-nonsense nanny. But the kids write their own ad, calling for a sweet and fun nanny. George tears this up and chucks it in the fireplace. But it gets blown up the chimney, and Mary Poppins magically gets a hold of it.
Even though Mary has a weird habit of reassembling thrown out garbage (like the ad), she's a nice person with magical powers. She arrives at the Banks house, zooming in on her flying umbrella as the wind blows George's strict nannies away. Flabbergasted, George has no choice but to hire her. She immediately astonishes Jane and Michael with her magical hijinks—helping them clean the nursery with magic that makes the nursery clean itself up. She also teaches them—and the rest of the world—that "A spoonful of sugar helps the medicine go down."
Next, she takes them through a series of adventures, first entering a cartoon world by jumping into Bert the cockney chimney sweep's chalk drawing on the sidewalk. (Bert and Mary seem to be old acquaintances, ambivalently attracted to each other…something's simmering on the burner, but it's hard to say what). After this frolic—involving the classic song "Supercalifragilisticexpialidocious"—they visit her Uncle Albert, who floats in the air when he laughs, and Mary sings them a song about an old lady who feeds pigeons.
Mary cheers everyone in the house up—except for Mr. Banks, who's obviously jealous of her popularity. He wants to fire her, but Mary prevents him by suggesting the children go to the bank and learn about Bank's lessons of discipline and thrift. However, this doesn't go so well: Michael gets into a tussle with the head of the bank, who tries to force Michael to invest his tuppence in a savings account. People at the bank misunderstand what's going on and all try to withdraw their money, in a panic.
After Michael, Jane, and Mary go on a song and dance excursion with Bert and his chimney sweep pals, Mr. Banks goes to the bank and gets fired because of Michael's behavior. Funnily enough, he finds he's totally excited to leave the bank and goes home feeling "Supercalifragilisticexpialidocious."
The next day, George arrives home a changed man, ready to have fun with his kids. He takes them out to fly a kite, and Mrs. Banks comes too. At the park, they run into the son of the deceased head of the bank, who gives George his job back—with a promotion to partner. At the same time, the wind changes, which means Mary Poppins needs to leave—her work there is done. Naturally, the kids are sad, but they're pumped up to have their dad happy and in a decent frame of mind.
Mary sails away on her umbrella, leaving a joyful Banks' family behind.