This is the most simplistic, straightforward and obvious way of voicing a short story. Your narrator is a character directly or tangentially involved in the story that relates his own experience and shapes the flow of the plot and details. Imagine telling someone an anecdote or an experience from your everyday life to get an idea of the approach. The popular American novels "The Great Gatsby" and "The Catcher in the Rye" are two examples of first-person writing. This style, along with third-person limited, are the two most common approaches used today.
This is an older approach that creates a third-person narrator who is not involved in the story and is sort of a disembodied, unnamed voice. The narrator is privy to the thoughts of all the characters in the story (hence "omniscient") and slides in and out of characters' heads as the plot warrants it. Virginia Woolf, William Faulkner and many 19th-century writers are famous for using this approach that allows the reader direct access to the psyche of all the characters of a story.
Third-person limited, as is implied in its name, is a variation on the removed third-person narrator that allows the reader only limited access to the thoughts of certain choice characters within the story. Typically the main character(s) and a supporting character are focused on rather than the aforementioned omniscient approach. This style allows more of a realistic approach while still utilizing the unnamed and uninvolved narrator. Both third-person approaches present a degree of objectivity in the storyteller that is not present in the first person.
Beyond just voicing options there are also word and style choices that create a story form in which to place your plot, setting and characters. Minimalism is the celebrated style of Ernest Hemingway and adherents who practice an economy of words in their writing. Less is more is an apt description, because the author often leaves a fair amount up to the reader to fill. The opposite is maximalism practiced by writers like Henry James who employ wordy descriptions and digressions throughout their work attempting to detail and explain as much as possible.