The movie begins with a vignette showing the planets Mars and Earth, during which the narrator explains that, in the summer of 2010, the planets came so close to each other that "a cosmic force was disturbed" and a baby destined for Earth arrived at Mars, and vice versa. Two comet-like bodies are shown colliding and then assuming paths distinct from their original directions of travel. The transit of one, colored green, is followed as it flies through Earth's atmosphere, above hundreds of homes with strange-looking TV antennas, then arrives at a high-rise hospital.
Wanda is waiting with other anxious, reading a newspaper in the hospital waiting room. Finally, an announcement comes over the P.A. that Wanda can see her baby. Excited, she presses against the glass of the nursery window while her baby is rolled in on a small gurney. The baby becomes visible, but wait! His head is green! Then, he jumps up from the gurney and we see that his head has two antennae that spark and make Morse-code style beeps! Wanda faints.
The next scene opens in a suburban neighborhood. Wanda is arguing with Cosmo. She's pleading to him to let the baby stay in the house. He counters that the baby needs sunshine and fresh air. So, Wanda pushes the baby in a stroller, fearful of being seen. While she's not looking, the baby crawls up on the stroller hood and beeps at Wanda, startling her. Then, the baby crawls up on a wall and communicates with a bee sitting on a nearby flower. We next see Wanda back at the house, pleading once again, unsuccessfully, to keep the baby home. Once again, she's pushing the stroller along when an elderly woman begins to dote on the baby, picking him up and noting that he's "such a healthy shade of green." As she begins to realize something's strange, the baby beeps his antennae at her and steals her glasses. Horrified, Wanda hurries the baby back to the house. The elderly lady, still unusually calm, pulls a tuning reed out of her pocket and uses it before letting out two bloodcurdling screams.
In the next scene, Cosmo is beginning to worry about the baby. He's doing the family's income taxes, spelling out Einstein's Mass–energy equivalence with letter blocks, and creating a Tinkertoy (named "Stinkertoy" in the cartoon) model of the (fictional) illudium molecule made famous in the Marvin the Martian cartoons. We're also shown a model of the Solar System made from a basketball and Christmas ornaments hung from the ceiling with string, and a graph on a chalkboard titled "Hurricane Possibilities for Year 2015." We see not only plans to build a better mousetrap, but corresponding blueprints on how to build a better mouse. Agreeing that "he should play more," Wanda sits the baby in front of the TV, where "Captain Denzel Crocker" is displaying a toy flying saucer being offered as a promotion for Cosmic Crunchies (even though the screen identifies it as "Ghastlies"), the "new wonder cereal made from unborn sweet peas." The baby brings in a T-square and triangle, measures the dimensions of the saucer displayed on the TV screen, and retires to his room, where he builds "his own toy spaceship."
Next, the family receives a letter from Mars delivered by a small rocket. The message, from "Sir U. Tan of Mars" (a reference to a popular vegetable laxative, "Serutan"), restates the event portrayed in the opening scene, adding that the Martian baby's name is "Mot." Furthermore, Tan states that their baby is on Mars and they call him "Poof." The Fairylings are cautioned to guard the baby carefully until an exchange can be made. At that moment, using his high chair for a launch pad, Mot launches in his "toy" spaceship out the bedroom window. The frightened Wanda first chases him by foot, then by car. Wanda reaches a high-rise hotel just as Mot is flying into a window on an upper floor. Inside the "Auditorium", a UFO skeptic is deriding the concept of "green men" and "flying saucers" until the little green baby in his flying saucer stops right in front of him, after which the skeptic bursts into tears.
Wanda arrives just as Mot is flying out another open window. She tries, unsuccessfully, to grab the spaceship, after which she falls out the window. Mot flies up to a waiting mother ship, which takes him in. The view then changes to Wanda screaming as she falls to oblivion and the street far below.
The scene fades and wavers to the P.A. in the hospital waiting room, where Wanda, this time alone in the waiting room, is called to see her baby. She'd apparently fallen asleep while reading a science magazine with the lead story of "Can we communicate with Mars?" Excited, she goes to the window of the nursery, this time to see a healthy fairy boy rolled in. She whistles with relief. The view, however, zooms in to the baby's wrist, where a bracelet is worn with the letters, "POOF."