In this context, a story is a particular business need assigned to the software development team. Using estimations of story points rather than time allows development teams to be less precise. It may be difficult, for example, to estimate how long a particular feature will take to develop but relatively easy to understand if it is more complex than others, in which case it should be assigned more story points.
Elements considered in assigning a story point include the complexity of the story, the number of unknown factors and the potential effort required to implement it. Story points are usually expressed according to a numerical range, such as an adaptation of a Fibonacci sequence, or according to a size range from X-S (extra-small) to X-L (extra large).
See a video introduction to story points: