Meemaw is Sheldon's nickname for his maternal grandmother, whose real name is unknown. "Meemaw" is Mary Cooper's mother, and Sheldon's favorite of all his family members as he shows her no resentment at all, unlike his resentment toward his other family members.
Through Sheldon's descriptions, Meemaw has made cookies for him and had him polish her silver. In turn for calling her Meemaw, Meemaw's nickname for Sheldon is "Moonpie", and according to Sheldon, no one can call him by that name except Meemaw.
Sheldon is known to receive and write letters to Meemaw, which Penny finds when he asks her to go into his room to find a flash-drive that he forgot to take with him during a trip.
His Meemaw is effectively Sheldon's weakness in several episodes. He generally refuses to consider any ill will or imperfection she has, including the thought that Meemaw needed to have sex in order to conceive his mother. Wil Wheaton also uses Sheldon's adoration for Meemaw against him, stating that his own grandmother passed away, which caused him to miss a convention of which Sheldon had attended to meet Wil Wheaton in his childhood. This story forces Sheldon to back down in a Mystic Warlords of Ka'a card tournament, relinquishing the championship title to Wheaton in the episode "The Creepy Candy Coating Corollary". This story was a ruse in order for Wheaton to win against Sheldon, of whom he couldn't beat through gameplay.
"Meemaw", like "mamaw", "mimah" and various other Southern US expressions for grandmother or mother, ultimately derives from Arcadian (Cajun) Louisiana French, in which the French maman ('mother') is prounounced more or less as "m'maw".