From the Super Mario Wiki
Grandma Toadstool (or Gramma) is the grandmother of Princess Toadstool. She first appeared on The Super Mario Bros. Super Show!. She is presumably the mother of the Mushroom King, with Toad being her grandson.
The Super Mario Bros. Super Show!
Prior to the events of the episode, Grandma had fallen ill, prompting Princess Toadstool to pay her a visit, along with Mario, Luigi and Toad. Due to the fact that King Koopa had stolen all the firewood from the local village, Mario, Luigi and Toad stayed behind to gather more, while Princess Toadstool ventured on to visit Grandma.
Meanwhile, Gramma's firewood was running low, when King Koopa, dressed as a messenger, tricked her into thinking that she had won a year's supply of wood, and convinced her to go into town to claim it, so that he could dress up as her and trick Princess Toadstool once she arrived. King Koopa's plan nearly worked, although Princess Toadstool escape from the cottage while Pine explained to Gramma that everyone's firewood had been stolen. Mario, Luigi and Toad told Gramma to stay behind while they went off to rescue Princess Toadstool, but their absence allowed for three Koopa Troopas to kidnap her.
After King Koopa kidnapped Princess Toadstool, he and his Koopa Troopas brought her and Gramma to his lair in a nearby cave, where he had been hoarding everyone's firewood. However, Mario, Luigi and Toad wore the Big Bad Wolf's spare winter coats to scare off King Koopa, as the Big Bad Wolf wanted to get back at him for fighting with him earlier on.
With everyone's firewood restored, Pine explained that everyone would be able to keep warm for the rest of the season. Gramma then thanked Princess for visiting, and suggested that maybe next time, she should come and visit her instead. Princess Toadstool responded by giving Gramma a kiss.