An upscale clothing shop located in Diagon Alley. After encountering Harry, Ron, and ‘mudblood’ Hermione in Madam Malkin’s, Narcissa and Draco Malfoy took their business to Twilfitt and Tattings instead (HBP6, Pm).
"Twilfitt" suggests the phrase "It will fit." Twill is a type of fabric woven with diagonal ridges.
"Tatting" = a kind of knotted, handmade lace used as trim for clothing. I can't help but also associate the word with the Cockney slang term "tat" which refers to the kind of junk souvenirs hawked to naive tourists in London.