Frances was the mother of Lily Buffay and the grandmother of Phoebe and Ursula Buffay.
Quick Answers
Who is Frances' biological granddaughter in 'Friends'?
What is the relationship between Frances and Phoebe Buffay?
Who was Lily Buffay's best friend?
How does Frances' character impact Phoebe's life in 'Friends'?
In which season of 'Friends' does Frances pass away?
Biography
Frances was Phoebe's grandmother, although they are not biologically related, as her daughter isn't the biological mother of Phoebe. Phoebe does not discover this until "The One At The Beach."
Until her death in Season 5, Frances and Phoebe lived together in her apartment. The two of them were very close, and Phoebe regularly talks about her grandmother with her friends. Frances does not appear to be as close with her other granddaughter, Ursula, who thought that Frances had passed away five years prior to her actual death.
Season 2
Frances' only physical appearance is in "The One With Phoebe's Dad," where Phoebe finds out that she has been lied to about what what her father looks like. All pictures that Phoebe have seen of her father, is actually just some random guy that came with the frame. Frances claim that it was Lily's idea, and that she just went along with it. She then gives Phoebe a real photo of her father, as well as his address, but Phoebe only finds her father's second wife and her younger half-brother, Frank Jr., who were left by Frank Buffay Sr. 4 years ago.
Frances also keeps a framed picture of Albert Einstein in her apartment, which Phoebe belives is a picture of her grandfather.
Season 5
In "The One With Joey's Bag", it is revealed that Frances had passed away while buying groceries at the market with Phoebe. As Phobe puts it, "she bent down to get some yogurt and she just never came back up". Although Phoebe loved Frances, she is sure that her grandmother is in hell. Phoebe informs Ursula about their grandmother's death, but Ursula declines the invitation to the funeral. Beliving that Frances had died five years earlier, Ursula had already made peace with their grandmother's death - plus, she had a concert to go to the same day.
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 with Ross) for the better part of a year after fire damage requires it 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.