Disclaimer: While this plot synopsis is for the benefit of anyone wishing to reminisce about the film, no possible summary could do its comedy justice.
Ted Striker is a ex-fighter pilot who flew in "The War"  but now has severe self-esteem issues due to a failed mission in which he lost most of his squadron. He drifts from job to job and is seen driving a taxi (badly) when the movie opens. He arrives at the airport in an attempt to stop his girlfriend, Elaine, an airline stewardess, before she leaves him for good. Elaine cites his lack of self-confidence and inability to get over the past as her reasons for dumping him, and is headed for Chicago to "start her life all over again".
Overcoming his fear of flying, Ted purchases a ticket for her flight and embarks to try to win her back. While aboard, he reminisces about his past with Elaine to a number of fellow passengers, who commit suicide from boredom.
As the flight progresses, the passengers and crew begin to fall ill. As luck would have it, there's a physician on board, Dr. Rumack, who diagnoses the illness as food poisoning from contaminated fish. As the pilots pass out, one by one, Elaine desperately calls Chicago for help and is told to activate the automatic pilot.
A surreptitious search then begins among the passengers for someone with enough flying experience to land the plane, as while Otto the automatic pilot can fly the plane, he can't land it. Naturally, Ted gets picked for this role, despite his protests, since he's the only one with any experience at flying. McCroskey, the Chicago air traffic controller, decides to get Rex Kramer, a veteran pilot, to talk Ted down.
As it turns out, Kramer was Striker's commander during the war, and the dislike is mutual. One microphone faux pas by Kramer later, Ted's in a Ten-Minute Retirement, and it falls to Dr. Rumack to talk him out of it with a Rousing Speech. Inspired, Striker returns to the cockpit and takes charge, determined to get the plane on the ground despite thick fog.
Naturally, he succeeds, in an epic crash landing, and leaves the plane hand in hand with Elaine, who has rediscovered her love for her man. The movie closes with Ted's taxi passenger (from his introductory scene) saying, "I'll give him fifteen more minutes, but that's it!"
- Implied by flashbacks to be World War Two but it couldn't possibly have been that because he's too young.