"The Family" is a show in the world of 'Fahrenheit 451' which is displayed on walls (I think Montag had recently bought his wife a third wall for her TV room and was bugging him for a fourth). It is basically a soap opera to the Nth degree. It is interactive to some extent and has a story that pushes it along but is often either unimportant or unintelligible but yet she sits in her room watching/talking/whatever with the fake people all day every day. . . not unlike those who are addicted to games and forums online that they use to replace a proper social life.
I figured it would be a pretty easy reference, I had to read it in HS and then later in English 1A and it was on several recommended reading lists from middle school up. The US school system fails again I guess.