Frances is the mother of Lily Buffay and the grandmother of Phoebe and Ursula Buffay. She was thought to be Phoebe's biological grandma, but Phoebe later learns that her birth mother is another woman named Phoebe Abbott.
Frances is Phoebe's adoptive grandmother by her adoptive mother Lily. Her only appearance takes place in "The One With Phoebe's Dad", where she reveals to Phoebe that the person in the pictures she keeps around the house is not Phoebe's father. It was Lily's idea to tell Phoebe the picture of the man that came with the frame was her father, and Frances just went along. She then gives Phoebe her real father's photo and address. At the revelation, Phoebe tries to find her father, Frank Buffay Sr., but only manages to find her father's second wife and her half-brother, Frank Jr., as her father has left them for 4 years.
Frances is very close to Phoebe. She is mentioned by Phoebe from time to time. They live together for years until Frances's death in season 5.
Frances keeps a framed picture of Albert Einstein in her apartment, whom Phoebe believes was her grandfather.
In "The One With Joey's Bag", we learn that Frances has passed away. While she was in the market with Phoebe, she, as Phoebe puts it, "bent down to get some yogurt and she just never came back up". The last thing Frances said to Phoebe was "You go get the eggs, I'll get the yogurt and we'll meet at the check out counter", which Phoebe firmly believes they eventually will. Phoebe says that, although she loved her grandmother and her grandmother loved her, Frances is in Hell.
Phoebe goes to inform Ursula about their grandmother's death, but Ursula turns down attending the funeral as she thought Frances died 5 years ago and she had already made peace with it; plus, she has a concert to go to.
At Frances's funeral, Phoebe meets her long gone father, Frank Buffay Sr.
Phoebe inherits her grandmother's yellow taxi and apartment. Frances' ashes were kept in a lock box in the cab that Joey happened to find on his way to Las Vegas. Phoebe keeps the taxi until the very end of the series. She also lives in the apartment even after her marriage to Mike, although she's forced to live with Monica and Chandler (and briefly lives with Ross) for the better part of a year after fire damage requires the flat to be rebuilt.
At one point in Season 7, Monica asks Phoebe for Frances' cookie recipe, which was burned in her apartment fire. It turns out that Frances stole the recipe from Nestle Tollhouse, prompting Phoebe to yell at the ground that this kind of thing is why her grandmother is in Hell.