Unsure what to do, the Tanners take ALF into their home and hide him from the Alien Task Force (a part of the U.S. military) and their nosy neighbors Trevor and Racquel Ochmonek (John LaMotta and Liz Sheridan), until he can repair his spacecraft. He generally hid in the kitchen. It was eventually revealed that ALF's home planet, Melmac, exploded because of a catastrophe involving nuclear bombs. In episode four of season one ALF tries to convince the president of the USA to stop the nuclear program as he is afraid that Earth might share Melmac's fate. ALF was off the planet because he was part of the Melmac Orbit Guard. Gordon Shumway (AKA ALF) is homeless, but he isn't the last survivor of his species. He became a permanent member of the family, although his culture shock, survivor guilt, general boredom, despair, and loneliness frequently caused difficulty for the Tanners.

While most of the science fiction of ALF was played for comedic value, there were a few references to actual topics in space exploration, for example ALF's using a radio signal as a beacon in the pilot episode. In the episode "Weird Science", ALF told Brian, who was building a model of the solar system for school, that there were two planets beyond Pluto called "Alvin" and "Dave". However, after a call was made to an astronomical organization, Willie explained that "Dave" could have been the planetoid Chiron, or "Object Kowal", after its discoverer.

The original series spans over four seasons and 102 episodes (each episode's name is also the name of a song relevant to the episode's plot), in which ALF learns about Earth culture and makes new friends both within and without the Tanner family, including Willie's brother Neal (Jim J. Bullock), Kate's mother Dorothy (Anne Meara) (with whom ALF has a love-hate relationship — he refers to her as the Wicked Witch of the West or the Witch of Endor), her husband Whizzer (Paul Dooley), the Ochmoneks' nephew Jake, a psychologist named Larry (Bill Daily), and a blind woman named Jody (Andrea Covell) (who never quite figures out that ALF isn't human, though she is aware through touch that he is short and very hairy). Changes pass within the Tanner household over the course of the series, including the birth of a new child, Eric (The explanation for adding a baby in the series was that Anne Schedeen was pregnant at the time); ALF's move from his initial quarters in the laundry room to a converted attic "apartment", and the death of Lucky the cat; in the final instance, ALF finds that, despite his occasional attempts to catch Lucky with the intention of making the cat a meal, he has come to love and respect the family pet too much to do anything untoward with Lucky's remains.

In the series finale, ALF is about to be rescued by other survivors of his home planet, but is instead captured by the American military, and the viewer is left to ponder ALF's ultimate fate. This was not supposed to be the finale, as the original airing ended on a "To Be Continued" note. The producers supposedly had a verbal agreement with NBC to get at least one more episode to resolve the cliffhanger. NBC never made good on the deal, and the series was cancelled. However, the story is concluded in the TV-movie Project ALF.