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The Large Family is a family with many children. (We're not talking Duggar-style large, but still pretty large.) Sara doesn't meet them until she's poor, and at that point they represent everything she's doesn't have: family, money, comfort, ease, and joy.
In the evening, when she passed houses whose windows were lighted up, she used to look into the warm rooms and amuse herself by imagining things about the people… The one she liked best she called the Large Family. (10.2)
Sara never met a situation she couldn't tell a story about, and the Large Family is no different: she makes up names and back stories about them. When she finally meets them, they pretty much live up to her expectations: Mrs. Carmichael is "pretty" and "comfortable," and she was "warm arms" (18.3). Naturally, such a nice lady was funny, friendly kids—and we get the idea that Sara is going to have some nice new friends to replace the stuffy ones she made at Miss Minchin's.