As soon as the author of a short story writes the story down, she has a copyright in the work. The author has the exclusive right to sell, publish, distribute, share and create derivative works from her story. The author can register her copyright with the United States Copyright Office; but she is not required to register to have a full copyright over the work. A copyright does not last forever, but eventually expires. When the copyright expires depends on several factors, including how the work was produced, when it was written and whether the identity of the author is known. Once a copyright expires, the story falls into the public domain for anyone to use.
An adaptation is considered a derivative work that the copyright holder has the exclusive right to make. Section 101 of Title 17 of the U.S. Code defines "derivative work" in part as "a work based upon one or more preexisting works, such as a ... dramatization ... motion picture version ... or any other form in which a work may be recast, transformed, or adapted." Thus, the author of a copyrighted short story has the right to adapt her story into another medium.
Adapting the Story
Just because the author of a short story has the exclusive right to make adaptations of her work does not necessarily mean that you cannot make an adaptation. If the work has passed into the public domain, you can adapt it as you wish. For some stories, you may need to wait a long time before an adaptation can be made. For example, a story written after January 1, 1978 most likely will not enter the public domain until 75 years after the author's death; a story written in 1754 passed into the public domain a long time ago. The author can also grant you permission to create an adaption. The author can charge you a fee for permission or deny permission completely.
If the author of the short story registers her work with the U.S. Copyright Office, she can sue you for copyright infringement if you adapt her story without permission. In some cases, she can seek either statutory damages or actual damages she incurred as the result of your copyright infringement. As a practical matter, you will most likely not be able to publish, produce or perform any adaption without permission from the author. Any publishers, producers or theater owners that distribute or show your work would also be infringing copyright. They will require you to have the author's permission before they have anything to do with your adaptation.