Writing uses several terms that can be applied to both general and specific things. In the case of the words "story" and "narrative," both can be used to describe something similar in a general sense, but something narrower when being specific. Basically, a story is what happens in a narrative. Narrative is how a story is told. As with most writing, the main factor in understanding the difference between a story and a narrative comes through context.
On a broad sense, the word "story" describes the overall arc of a set of scenes sequenced to seem significant. It can be something as simple as relating a personal anecdote, with a beginning, middle and end. Meanwhile narrative can be defined as a flow of events used to relate a certain theme. The way the personal anecdote is told, what aspects of the story are emphasized and the words used to relate it are all features of the narrative.
The word "story" is often used synonymously with the word "plot," although plot is generally limited to where a story begins. Every story must have a starting point, some event or occurrence that serves as the driving force for the action. This could even be as simple as "I was standing in line at the supermarket the other day and this woman says to me ..." Then the story takes over and a series of events that occur following the established plot.
Point of View
The word narrative is also used to describe the point of view from which the story is told. A first-person narrative uses the "I" pronoun, where the narrator (speaker) is telling the story on her own. Second person uses the "you" pronoun; the narrator is telling what happens to "you." Third person uses "he," "she," "they," "it" and so forth. Third person allows both the most distance, by staying out of the thoughts of the characters, and the most access, by jumping into the thoughts of any character, depending on the limits of the narration.
Story and narrative are pieces of the each other. A series of thematic narratives, whether anything actually happens in them or not -- for example first-person thoughts while standing in line at the supermarket -- can be assembled into a story by placing them together in significant ways, perhaps by showing how the narrator's relationships change over time as demonstrating by who and what he or she thinks about while standing in line. At the same time, a series of story scenes can be assembled to create an overall narrative. Several stories on different encounters in supermarket lines can be placed together to form a narrative on socialization between strangers within a common setting. Here the narrative would be overall theme, structure and telling, while the stories would be the individual encounters used to build that theme.