Flash fiction is considered to be any story of 1,000 words or less. This genre has a protagonist and conflict but might not resolve -- leaving many things before and after the action implied. There is a moment or an occurrence. Prose poetry does not confine itself to the restrictions of story in any way -- it is open to anything. Some consider anything from a couple hundred words all the way down to several to be "micro-fiction," but this depends on how many labels you wish to consider relevant.
A short story is a work of fiction. There is a narrator, and a protagonist takes readers through conflict and resolution -- these are the basic elements of story. Most fiction magazines will consider anything from 2,000 to 5,000 words a publishable short story, though some consider up to 10,000 words (but only if it needs every word of it).
In a short story, a character usually experiences some kind of change. Most or all of the story -- beginning, middle and end -- revolves around this change. Short stories do not contain an overabundance of characters or deviations from a central plot or theme, as a novel might. Yet, unlike flash-fiction, they give readers something focused to think about, rather than a simple act to interpret.
Short stories are often gathered into collections and sold as regular-length books. They look like ordinary novels of several-hundred pages in length. These collections may be from either one or many authors, and the stories might feature a theme or character in common. They do, however, work as individual tales, which qualifies them as short stories.