A narrative is a moving picture. Like description, narratives need to have a rich texture of details so that the reader is seeing, hearing, smelling, and touching. The reader should experience the story, not simply hear it.
Stories add the element of time to description. Often stories start at the beginning and then follow the sequence of events chronologically. However, an effective variation on this pattern is to start in the middle of things and then use flashbacks to fill in the background information. This method is especailly effective in holding the reader's attention.
There are two extremes you want to avoid in writing a narrative. First, you can simply tell the story, event by event, without giving it any texture because you leave out descriptive details and dialogue. At the opposite extreme is a narrative that attempts to tell everything, painting detailed descriptions of every scene, quoting everything that is said, even speculating about the thoughts of the characters. A good narrative has texture, but it is suggestive rather than exhaustive. After all, the reader's imagination needs some room to fill in details. Giving too many details not only overwhelms the reader's imagination, it also slows the pace of the narrative.
Pacing is an important concept in narrative writing. Basically, pacing means that the writer sometimes slows the pace by putting more detail in, but sometimes she also hurries over details. A good way to know where to put in details and where to leave them out is to think of a narrative as consisting of episodes (smaller scenes that are strung together to make up a longer story). If you divide your story into a few short episodes, then you want suggestive detail within the episodes, but you want to hurry over the transitions between them. Think of episodes as pearls on a string. Make the pearls full orbed; keep the string stringy. The reader dwells in the episodes, but she needs to be oriented to them, and that is the function of the transitions.
As with description, point of view is important. What position is the story being told from? Another way of talking about this is to talk about the story's narrator. The narrator is not the writer, but the consciousness through which the story is told. Sometimes the story is told in third person, which means that every one is referred to as he, or she, or they. Sometimes, however, it is told in first person, which means that the narrator refers to himself as "I" and is actually involved in the story. Not all narrators are reliable.
The more sophisticated narratives become, the more problematic is the narrator. When the narrator tells the story in first person, but details in the story lead the reader to suspect that the narrator is not reliable, the result is irony. Irony is a narrative condition in which the reader and the writer share a common judgmental attitude toward the narrator, or when the reader knows more than the narrator and characters in the story.
For this assignment, it is probably better to tell the story as straight as possible. Irony is hard to pull off successfully. If you want to experiment with narrative form, I would suggest that you start somewhere in the middle of things and then use flashbacks. Also work on putting in suggestive but not overwhelming detail and dialogue. Try dividing your story into short episodes that build on each other. If you can pattern a sequence of events so that the story has some kind of climax (a scene of great tension and even explosion) followed by a denoument (a scene in which everything is worked out), you will have done more than many of us can.