This is both a good and incredibly difficult question to answer fully. It could be the subject of multiple books. Indeed, it is. A good place to begin to look might be with the work of one of the great modern story-tellers. Stephen King's 'On Writing' gives some pretty straightforward advice on the basis of how he constructs some of the best-selling fiction of the current era and, if you're interested in powerful story-telling, Elmore Leonard writes tellingly about his craft too. However, I am going to give you what I think are a few of the basic elements that I think contribute to 'powerful' stories, a term I am going to understand as meaning stories that create great, immediate and lasting impact.
Perhaps the most telling means of immediately making an impression as a story-teller is via a powerful narrative voice. The style of this voice might be very different according to the purpose of the writer. For example, the first person narrator, Holden Caulfield, of J.D. Salinger's 'Catcher in the Rye' is immediately striking for the caustic, cynical and guarded nature of his perceptions and by his use of 1950s New York slang, something that made a powerful impression on critics at the time. The spare, clean prose of some of Hemingway's early short stories (I'd pick out 'Up in Michigan' or 'The Big Two-Hearted River' as examples) is equally unique.
Interesting narrative points of view also work well. Take, for example, 'The Curious Incident of the Dog in the Night Time' by Mark Haddon, which captures the unique perspective of a boy on the autistic spectrum and his unique view of life or, indeed, James Herbert's poignant and disturbing 'Fluke', narrated from the perspective of a dog, the main character of the novel, similar in this respect to the very different voice of Jack London who does the same thing in 'The Call of the Wild'.
How an author creates believable characters and setting are absolutely vital too. A good writer is one who can hone in on a few characteristics, elements of physical description, movement or the intonations of speech to give the reader a sense of the character so strong that they feel that they are alive in front of them. This is evidently done in different ways by different writers. Take, for example, the names that Dickens gives his characters. You could tell that Noddy Boffin is pleasant and bumbling or that Wackford Squeers is more sinister just by their names. Character nomination is a key facet of making a chacter come alive. It's not just true of characters either: take 'Warings' the name of the house in Susan Hill's 'I'm the King of the Castle' which makes the reader alert to the need to be aware in a manner that the central character, Kingshaw isn't as he becomes increasingly bullied and isolated by the other protagonist of that novel, Edmund Hooper.
Susan Hill's novel has great setting too, another part of what makes for powerful story-telling. One of the early and traumatic episodes of that novel comes in a scarlet painted room, the red room, where Kingship is locked by Hooper in a dark room full of dead moths, the hobby of Hooper's recently deceased and sinister grandfather. The creepy nature of the room is carefully constructed by Hill to give a sense of the cruelty, order and cold calculating nature that Hooper has inherited from his grandfather and the setting of the book is subtly used, not only to provide a background for the action but also to give subtle hints (a technique known as fore-shadowing) of what is to come later in the story.
Many writers manage to capture a sense of place wonderfully in their work and to bring it alive for the reader. Jack Kerouac, the author of the wonderful 'On the Road', was so enamoured of the sense of place that he used to regularly perform a form of creative writing exercise that he called 'sketching' in order to try to write freely and without stopping about the places he was in so as to hone his observation and ability to write in concrete detail. He used these 'sketches' to make the places in his stories come alive and, whatever one might think about the morality of his novel, it certainly gives a distinct and vivid picture of 1950s America.
I don't feel that I have done much more than scratched the surface of your question but hope that this is of some help. A couple of further books that might help you to find out more:
- The University of East Anglia Creative Writing Guide is an excellent resource that illustrates many exercises about powerful story-telling with examples from some of the best modern and classic writers as well as providing chances to try to write for oneself.
- AQA, a UK examination board, produce a textbook for their English Literature A AS Level specification, Unit 1 called 'Aspects of Narrative' which expands on much of what I've discussed here. It may be useful.
I love Ashley Kannon's response, and I would like to embellish on it a bit with the idea of "windows and mirrors," a phrase the origin of which I do not know whom to credit, sadly, and one other element that I think is important.
"Windows and mirrors" is a phrase I heard repeatedly in graduate school, which is a metaphor for two ideas: the idea that powerful literature should give us a glimpse into a world other than our own or give us a mirror in which we can see ourselves, gaining personal insight. And of course, literature can do both, as well, in the same work, as for example, The Great Gatsby (Fitzgerald) does, allowing us to see our personal yearning selves, the world of the very wealthy and the very poor, and how one era of our history reflects our striving as a nation. A powerful story, either way, is a means of building empathy in us, enabling us to know ourselves better, which I believe makes us more likely to be empathetic, and which enables us to know others better, which is another good starting point for empathy. There have been studies done in which it has been established that those who read literature are far more empathic than those who do not read. Those of us who read and those of us who teach were not particularly surprised by these results.
On another note, as someone who has critiqued thousands of examples of student writing over the years, as well as read books too innumerable to count, I have come to believe that the selection of details that fashions a story renders it powerful or powerless. Have you ever listened to or read a boring story? What often makes it boring is the details that do not contribute to or sustain the point of the story, details that the storyteller should have left out because they slow down its thrust, because they do not make its point, or because some things the reader should have to imagine him or herself. Life is not a story; it happens in ways that are too messy, too slow, too fast, or too boring. What makes us powerful storytellers is the ability to impose a narrative that has a point, and the way to do that is in the selection of details that contribute in some way to that point. Another good example of this is from The Great Gatsby, in which there are lengthy passages describing what is served at the parties, the people who are present, or even all of Gatsby's shirts. These are purposeful details, meant to advance the reader's understanding of Gatsby and his yearning. And yet, all that Fitzgerald described of Gatsby physically was his smile. This was a purposeful omission of details, allowing each reader to see Gatsby his or her own way. In short, details matter, and which details matter a great deal.
There will be many different answers to this question because readers will view the word "powerful" in different ways. In my mind, "powerful" is the ability to illuminate a believable conflict between characters. This conflict is powerful when it is real life and something with which I can empathize. The most powerful stories and storytellers share a characteristic of empathy, expanding my moral and ethical imagination with what is happening in a particular story. For example, Hemingway's "Hills Like White Elephants" is a powerful story to me because it places me in a conflict between two characters. It is powerful because the story triggers a sense of empathy between both characters. Certainly, I might able to side with one character over the other, but in being able to place me in the middle of a conflict between two people, one that is real and distinct, I find the story and storytelling compelling. It is here where I find that one characteristic of the most powerful stories and storytellers emerges.
Another trait of powerful story and storytellers is the ability to illuminate some aspect of the human predicament. Morrison's Beloved is one such example of how the human predicament through literature can represent powerful stories and powerful story tellers. Sethe's predicament of struggling with the past, present, and future is a human one. A reason why the story is powerful and the way it is told is powerful is because it illuminates a component of what it means to be human. While the story is about the horrors of of slavery, there is a uniquely human dimension to what Sethe experiences. Injustice plagues human beings as a part of their lives and the struggle to understand and potentially overcome it in the name of having "more tomorrows" than yesterdays is a particularly powerful element. The illumination of characterizations where the human predicament is on display represents a characteristic of the most powerful stories and storytellers.
The characteristic of "Power" in literature represents how the author and story accomplishes the E.M. Forster dictum of "only connect." The ability to connect through literature is a distinct characteristic of the most powerful stories and storytellers. When this connection is forged, literature moves past something static and lifeless, and articulates what it means to be a human being and represents part of our own identity.
For me, the qualities that most characterize a "power" story or storyteller are believe-ability and memorability.
The more believable a story is, the more powerful it is because it slightly transcends the boundary between fiction and reality. Stories "come to life" when they include intricate details and paint a picture of what happens through its diction, imagery, and other figurative language. Storytellers have a hand in this aspect through their delivery methods, such as when they pause slightly or the inflection of their voices.
If you think about memories instead of stories, we say that memories that are easily recalled are very powerful. The same logic can be applied to stories - if you remember them for a long time, it must have some powerful element, like if the subject matter was very personally significant.