It is July 4, 1978. David Freeman (Joey Cramer) is an ordinary 12-year-old boy living in Fort Lauderdale, Florida, in an ordinary family, (Cliff De Young, Veronica Cartwright, Albie Whitaker), forever fighting with his little brother, Jeff (Whitaker), and beginning to discover girls for the first time.
Sent by his mother, Helen (Cartwright), to rendezvous with his brother, David falls into a ravine and is knocked unconscious. He wakes up and returns home to find that his family no longer lives there. Fort Lauderdale police reunite him with his parents and brother, but they have aged, and regard him as having been missing for 8 years. Elsewhere, an alien spacecraft crashes into an electrical tower and is rendered seemingly inert. NASA agents, led by Dr. Louis Faraday (Howard Hesseman), confiscate the vehicle, explaining to police that it is their experimental spacelab, and bring it to their own hangar, but are unable to penetrate it.
David is taken to hospital for testing and observation, and learns that eight years have passed. During the testing, David's mind exhibits images of the alien spacecraft, and the lead doctor alerts NASA. Faraday convinces the boy and his family that the truth will be discovered concerning his absence within 48 hours, takes him to his research facility, and locks him away as a "national security risk".
Faraday runs his own tests on David, and finds the boy's mind overflowing with alien technical manuals and also star charts covering expanses of the galaxy far exceeding anything Earth observers have recorded. David's subconscious mind tells the scientists that he was taken to a planet called Phaelon, 560 light-years away, in just over a two-hour journey, and the scientists realize that he has been subjected to time dilation as a result of having travelled at speeds far exceeding light, thus explaining why eight years have passed on Earth, but not for David. David himself, unable to comprehend what Faraday is telling him, panics and flees the room, leaving Faraday muttering that 48 hours will be insufficient to finish his investigation. An intern, Carolyn McAdams, (Sarah Jessica Parker) charged with David's upkeep informs him, and the boy tells her to alert his family.
David escapes by hiding in a robotic vehicle, which takes him to the hangar where the ship is being held. David boards the ship, and its robotic commander, (voiced by Paul Reubens, credited as Paul Mall), referring to David as "Navigator", evades NASA pursuit after some initial confusion and hides on the ocean floor. The commander, who refers to himself as a "Trimaxion Drone Ship" and is addressed by David as "Max", tells David that his mission was to travel the galaxy, collect biological specimens, take them to Phaelon for analysis and then return them to their homes. Phaelon's scientists discovered humans only use 10% of their brain and, as an experiment, filled the remainder of David's brain with miscellaneous information, including all of the star charts. Before leaving Earth, Max accidentally crashed the ship into the power structures, erasing all the computer's star charts and data. He therefore needs the information in David's brain to return home. Max then returned him to Earth, but didn't take him back to his own time, fearing that humans would have been vapourised during the process. David is shown the eight remaining alien specimens on board, and bonds with the Puckmaren, a tiny Bat like alien species and last of its kind.
Max performs a brain scan on David to reacquire the star charts, but in the process also contracts human emotional attributes. David and Max start bickering, and Max shuts down and allows the ship to fall from the sky, taunting David as he struggles to gain control. Eventually controlling the ship, David wanders the planet looking for Fort Lauderdale, triggering several UFO reports. In the meantime, Carolyn, the intern, has made contact with David's family and told them about his escape in the ship; as a result, Dr. Faraday has them put under house arrest. David arrives in the Florida Keys, and telephones his family to ask Jeff to signal from the new house so that he may find it. Jeff sets off the fireworks that had been intended for use in 1978, but of course never were set off. David homes in on the fireworks, but NASA agents, having tracked the ship's every move, are there first. David, realizing what a circus his life would be if he remained in 1986, orders Max to return him to 1978, regardless of the risk to his life.
The time travel was a success and David wakes up in the ravine he fell into, walks home, and finds everything as he left it. On the family boat, Jeff sees Puckmaren in David's pack, and David spots Max in the sky who hears him yell "see ya later, navigator" before vanishing into the sky.