"The family" is a reference to the television programs that Mildred, her friends, and most of Montag's society are so addicted to. Montag jokes around and calls these characters the family, because Mildred is so emotionally invested in their fake lives, and spends much more time with them than with any actual, real family. Television has taken over the lives of many people in this society, turning them into drones that escape any form of reality through the made-up worlds of the television programs. To a certain extent, that happens today also. We become engrossed in our programs, watching them every week, becoming emotionally attached to the characters.
The strong role that television programs, and "the family" plays in this society is to deaden any independent thinking, and to give the people an outlet for intense emotions and ties, that is safe and non-rebellious to the government. If you have people that have free time to read books, to sit around and talk, to discuss politics and morals, then you have a society that is discontented with an oppressive government, and who will possibly rebel against it. Montag's society operates on the principle that the more you fill a person's head with nonsense, the less room there is in for rebellious, independent thought that might challenge the government's authority. The government likes power, and a people that is so dumbed down by their addiction to worthless entertainment is the perfect mass of sheep to control. "The family" keeps people's emotions vented so they don't get restless, it gives them a sense of meaning and purpose through fake relationships so that they aren't unhappy with their own empty relationships, and keeps them tuned out and unthinking. It serves the purpose to keep the people submissive to the government's control. I hope that those thoughts helped; good luck!