Narrative poetry tells a story. Like a short story, it has a beginning, a middle and an end. The beginning generally gives the story setting. The middle brings in the conflict, and the end resolves the conflict. However, a short story will usually offer more information such as what the character looks like, the setting and sensory details.
Narrative poems and short stories both tell a story about a main character, the hero, and his adventures. Most are told in the third person. Short stories also are often told from third-person point of view, though sometimes they are told in the first person. In both, the central character must face an event or challenge that has to be overcome.
In both the short story and narrative poem, the character must face challenges and overcome those challenges. The conflict in both leads to a climax and resolution where the character either learns a lesson, or simply overcomes the obstacles presented in the story. This is what makes the narrative poem different from a lyric poem and a short story different from an essay.
Hundreds of years ago, stories and poems were both passed down by word of mouth. Aesop had his moral fables and Homer his epic poems of heroic deeds. Both have stood the test of time and are still widely read today. Poems both long and short still have a place in today's market as do the short story and novella -- but modern narrative poems are extremely rare and little read.