Ocarina of Time, Granny's Potion Shop is located in the back of the Kakariko Village. Although it is never directly stated, Granny is either the mother or wife of the Carpenters' Boss. This would make her either the grandmother or mother of the Cucco Lady and Grog.
Granny's Potion Shop is only accessible by flying into its yard with a Cucco. The shop, however, is closed and a message is on the door letting passers-by know that Granny is out in the field collecting herbs for her potions. Seven years in the future, Link is able to gain access to her shop (either using the Hookshot from the windmill, or the back door of the Medicine Shop).
After Link delivers Cojiro to Grog in the Lost Woods he asks the young hero to deliver an Odd Mushroom to Granny. Upon showing the Odd Mushroom to the Potion Shop Lady, she calls Grog a "bum" for going into the forest, and asks Link that if he sees "that fool", to give him the Odd Potion. However, she says, the potion will not work on a monster, possibly hinting at Grog's fate as a Stalfos.
The Lake Scientist, who lives in the Lakeside Laboratory, mentions a mutual friendship with Granny, saying that she is like a student of him.
Majora's Mask
In Majora's Mask, Anju's Grandmother is Granny's Terminan counterpart. Anju's Grandmother lives in the lower bedroom of the Stock Pot Inn in Clock Town. While confined to her wheelchair, she serves as a storyteller. Unfortunately, her stories are unbearably long and the listener will often fall asleep in the middle of them. She also appears to be losing her memory, as she often refers to her granddaughter Anju and Link by the name of Tortus, her deceased son and Anju's father. On the final day, she flees to Romani Ranch with Anju and her mother.

Anju's Grandmother
The Bombers' Notebook
Anju's Grandmother is one of twenty characters in Majora's Mask who has an entry in the Bombers' Notebook. Anju's Grandmother's entry can be completed at anytime between 8am and 6pm on the first or second day.
In order to complete the entry, all that is required is to stay awake during each of her two stories and answer a question at the end. The reward is one Piece of Heart for each story. However, in order to stay awake during the two stories, Link will need to wear the All-Night Mask while listening to the tales.
The Stories
The first story, about the "Carnival of Time", is the shorter of the two and takes two hours to tell. At the end, Anju's Grandmother will ask when the Clock Tower opens. Link must answer "On the eve of the carnival" to receive a Piece of Heart.
The second story, about the "Four Giants", will take all of the time between when the story begins and the next morning. Thus, if the story begins at 6 am, it will not end until 6 am the next day. After listening to the story with the All-Night Mask, Anju's Grandmother will ask what the people do to call the Four Giants. Link must answer "I dunno" to receive another Piece of Heart.
Majora's Mask Manga

Non-Canon Information: The following article or section is not Zelda canon and shouldn't be taken as such.
In the Majora's Mask manga, Anju's Grandmother is depicted much the same as she is in the game, with yet another hint that she may be faking her senility. This is ignored by Anju's Mother, who swiftly forces her out the door as they make their way to Romani Ranch.