An alien expedition lands in Rutherford, Ohio, where they take on the form of what they think is a typical human family. Although only intending to stay for a couple of weeks, the aliens rent an apartment from carefree landlady Mrs. Dubcek, Dick acquires a job as a Professor of Physics at the nearby Pendelton State University, and Tommy is sent to school. However, Dick, upon discovering the concept of emotions and acting on his infatuation with Dr. Mary Albright, extends their mission on Earth.
The aliens become sick for the first time and jump to the conclusion that they are dying. Despite being unwell, Dick accompanies Mary to a wedding, where he tries, in vain, to hide his ill health from her. However, after drinking far too much cough syrup and making a scene both during the service and at the party, Dick is taken home by Mary, where, believing he doesn't have much time left, he tells her he loves her.
The aliens discover the concept of age and Dick chooses a birthday. However, after Mary gives him a card that jokes about him looking "like a million dollars... old and wrinkled", he becomes offended and struggles to come to terms with his being 'over the hill', and so strives to become young again, by dying his hair and wearing ridiculously tight leather trousers.
Sally has her very first date, but becomes upset when he doesn't call her back afterwards, so Dick asks Mary and Nina to take her out and drown her sorrows. Meanwhile, Tommy starts high school, and Dick tries to bond with him in a fatherly way.
When an old boyfriend of Mary's (Ed Begley, Jr.) makes a re-appearance, Dick experiences the human feeling of jealousy. Meanwhile, Harry adopts a stray dog, but Mrs. Dubcek is allergic to dogs and will not allow him to keep it in the house, so he gives his dog to a single mother Kate (Katherine LaNasa).
When Mary goes away on vacation, Dick misses her badly and, in desperation to find contact details for where she is staying, takes Nina out for dinner and plagues her for details; however, when this doesn't work, he breaks into Mary's house. Meanwhile, Sally and Harry become addicted to skin care products that promise them eternal youth, and Tommy meets a girl called August Leffler (Shay Astar), whom he begins to date.
During a party honoring him, an obnoxious, sexist professor (John Mahoney) dies, and his will reveals that he wants Dick to deliver a "forthright and honest" eulogy about him, despite the fact that he is a universally despised man. Meanwhile, Tommy neglects his girlfriend August in favor of hitting it off with an "easy" cheerleader.
Dick wants the family to get along and stop irritating each other, and thinks that they can learn family values from observing Dr. Albright and her loopy brother Roy (Bronson Pinchot), so he invites them to dinner. However, the Solomons panic when Roy claims to have previously been abducted by aliens. Meanwhile, August is annoyed with Tommy, because Dick made him cancel his date with her to have dinner with Mary and Roy.
Dick discovers the consequences of telling the truth when he repeats some of Dr. Albright's comments about how boring the academic committee are, at an academic committee meeting he attends. Predictably, Mary is annoyed, and enlightens Dick to the concept of 'bending the truth'. Meanwhile, Sally and Harry attempt to get a driver's license, but discover that it is more difficult that they'd expected.
Harry cannot seem to find anything he is good at, but after painting the side of Mrs. Dubcek's house, decides that he might be good at art, and takes a class in art. However, Dick becomes annoyed when Harry turns out to be much better at art than he is. Meanwhile, Sally, in a strong and dictatorial way, takes over the bake sale at Tommy's high school.
Mary and Dick are going to Chicago to be presented with awards for teaching, but upon getting on the plane, Dick panics and gets the pair thrown off. They begin to drive there but when it snows, Dick panics again, believing the flakes to be alien. He commandeers the car, but promptly crashes it, leaving the pair of them deserted at a truck stop. Meanwhile, Harry gets a job in a video rental store and takes it upon himself to rearrange all the videos according to his preference, while Sally, believing the human race to be dead, attempts to re-start mankind with Dick's student Leon, who arrives to drop off some homework.
Dick befriends his next door neighbor, while Sally befriends his wife, but when the Solomons drop by to pay a visit in the early hours in the morning, their friends become angry. Dick visits his friend to apologize and offer him a gift, but he becomes angry at trying to be bought, and the pair end up fighting.
Nina tells Dick that she can't find any graduate records on him, which causes him to panic and act defensively. This leads Mary to note that Dick is often secretive about his past, and, while reminiscing about her rebellious time at Berkeley in the sixties, conclude that Dick is actually Manny Rosenberg, an activist whom she briefly knew, who is in hiding from the FBI. Dick first denies this, but then realizes that his presumed identity may have made him more attractive to her. Meanwhile, Harry joins a CD club, but finds the responsibilities of membership overwhelming.
Dick starts to feel outcast from the others in his office, feeling that because he is male, he is excluded; he becomes especially annoyed when he can't come to Mary and Nina's women-only study group. Meanwhile, Sally and Harry get a job at a pancake house, where Sally becomes irritated at the lack of tips, and Tommy considers telling his girlfriend August the truth about where he and his family really come from.
When Nina arrives at work in an ethnic headdress, Dick begins to realize that the Solomons have no ethnic group, and takes it upon himself to research them all and decide which one is the "best". Meanwhile, Sally falls for Tommy's teacher, Mr. Randell, who invites her to attend the school dance with him; however, she becomes nervous when she realizes that she doesn't know how to dance.
After the radio is stolen from the Rambler and Dick gets mugged at an ATM, Sally begins to worry that she is failing in her job as Security Officer for the unit, and the family files a police report with officer Don (first appearance of Wayne Knight).
Dick decides that he is now Harry's father, and tries to treat him as a son, but Harry wants to know what his use on the alien's mission is, and Dick and tells him about the radio receiver in his head. Unfortunately, Harry becomes angry and runs away from home. Meanwhile, Sally goes on a date with Mr. Randall and meets his mother (Naomi Judd).
Finding out another professor is leaving and a better office is available, Mary is able to get it over an older professor (Harry Morgan); however, Dick misses her and schemes to get her back into the old office with him. Meanwhile, after fainting when Mrs. Dubcek cuts her finger, Sally ends up in the hospital, and the family ends up acting like doctors from every medical drama they have seen, until they are caught.
Mary finally agrees to a date with Dick, and eats out with him at a Japanese restaurant, where Dick shows off his skills at chopping up food. The pair then retire to bed and finally have sex for the first time. Bad news arrives, however, after Tommy's report to the Big Giant Head results in him deciding that they've wasted their six months on Earth, and he announces that a replacement for Dick will be sent shortly.