What is the difference between narrative prose and dialogue in a short story?
The term "prose" simply means anything other than poetry, so nearly every book or short story you have read is written in prose. (A few notable exceptions would be Seamus Heaney's translation of Beowulf and John Milton's Paradise Lost which are both written in poetry form, also known as verse.)
Adding the word "narrative" simply adds the component of storytelling to the prose work. So, narrative prose is a format used to tell stories. Narrative prose may contain some elements of poetry (such as figurative language and imagery) but will not be written using rhyme and meter, as the two works mentioned above.
Within a work of narrative prose, an author often includes dialogue, exact words spoken directly by the characters, rather than a paraphrased version given by a narrator. For example, the following short excerpt from William Golding's Lord of the Flies shows both narration and dialogue:
"Nobody knows where we are, " said Piggy. He was paler than before and breathless. "Perhaps they knew where we was going to; and perhaps not. But they don't know where we are 'cos we never got there. " He gaped at them for a moment, then swayed and sat down. Ralph took the conch from his hands.
This is a narrative prose work which includes dialogue. Nearly every short story and novel author uses both narration and dialogue to create the most interesting and compelling stories.