Please explain how the characters of Fahrenheit 451 describe family differently, and support your explanation with quotes and ideas.

For Mildred, family consists of the characters in her television shows. When Montag asks her to turn off the parlor walls on which she watches her shows, she responds, "That's my family."
It is telling that she doesn't consider Guy, her husband, her family, but instead feels closer to her television characters. Later, when Guy shuts off her parlor walls, she will explain to him that her television family:

tell[s] me things; I laugh, they laugh! And the colors!

She also fears that if Beatty finds out Montag has books he:

. . . might come and burn the house and the family. That's awful!

But then Montag asks her:

"Millie, does—" He licked his lips. "Does your family love you, love you very much, love you with all their heart and soul, Millie?"

Millie doesn't want to answer, calling it a "silly question." Of course, she knows in her heart that her television family doesn't care about her at all. Montag is her real family, but she doesn't recognize that and pushes him away. A televised family that she can't get into conflict with is easier for her to deal with than a husband with odd ideas.
Bradbury contrasts the sad emptiness of Mildred's concept of family with Clarisse's family. Her family, the McClellans, consists of real people who actually have conversations with her and each other. When she and Montag come to her home, he is surprised to see her house ablaze in lights and asks what is going on. She says:

Oh, just my mother and father and uncle sitting around, talking.

To Montag, this is very unusual. It is to Beatty too, who refers to the McClellans as the kind of "odd ducks" who need to be rooted out for the good of society.
In this upside-down world, real family ties are considered subversive and abnormal, while having a fake TV-family is the norm.