In "The Open Window," what kind of dialogue is spoken by Vera the niece?
The dialogue given to Vera, Mrs. Sappleton's niece, is direct dialogue, also called direct speech. At one point she embeds a narrative within her direct dialogue. Direct dialogue is established with Vera's opening lines at the beginning of the story:
"My aunt will be down presently, Mr. Nuttel," said a very self-possessed young lady of fifteen; "in the meantime you must try and put up with me."
It is while she is spinning her specialty of "[r]omance at short notice" that a narrative is embedded within her direct speech. To explain more, while Vera is speaking directly to Mr. Nuttell, she tells him a tale about Mrs. Sappleton's troubles related to the (imaginary) disappearance of her husband and brothers and their hunting dog.
In crossing the moor to their favourite snipe-shooting ground they were all three engulfed in a treacherous piece of bog. It had been that dreadful wet summer, you know, and places that were safe in other years gave way suddenly without warning.
Part of Mrs. Sappleton's conversation, however, is reported by the narrator indirectly: we are told what she has said without her saying it in quotation marks:
She rattled on cheerfully about the shooting and the scarcity of birds, and the prospects for duck in the winter.