What exactly is a story point? Answers to this question are all over the map. I know because I went looking.
Maybe a story point is a nebulous unit of time: “Story points represent nebulous units of time.”
Maybe a story point measures effort: “Story point is a random measure used by Scrum teams. This is used to measure the effort required to implement a story.” 
Does a story point measure size and complexity? “So, points are relative measurements of size/complexity, not absolute measurements of duration.”
Or perhaps a story point is many things: “…a story point estimate is an amalgamation of effort involved in developing the feature, the complexity of developing it, the risk inherent in it, and so on.”
How are story points used? Again, opinions differ wildly. Maybe the number of story points in a sprint (velocity) is a measure of productivity: “Truth be told velocity really is a productivity measure…”
Maybe story points should not be used to measure productivity: “Using story points or ideal days to measure productivity is a bad idea because it will lead the team to gradually inflate the meaning of a point…”
Maybe story points should be burned down during a sprint: “Basic Truths about HyperProductive Scrum…Burn down Story points only.”
Or maybe it’s the other way around and story points should not be burned down during a sprint: “I also recommend estimating the sprint backlog in hours rather than in points.”
When it comes to story points, it’s a wild rumpus of opinion. Some writers believe that teams achieve steady state velocity, others do not. Some believe that there is a mapping between story points and time, while others do not.
When I started writing this note, I hoped to answer the question in the title. After doing this research, I confess, I can’t tell any more what a story point is or what it is good for.
I now think that story points and associated practices (like burndown charts or poker planning) conflate so many intellectually distinct ideas and goals, from increasing shared understanding to avoiding gaming, that I am searching for a new way to describe them and introduce them to people who are new to Scrum.
The Scrum Guide does not mention story points at all — that may be the best way to go.
[An extended discussion of this essay appears here.]
 Sutherland, Jeff. “Practical Roadmap to Great Scrum:
Systematically Achieving Hyperproductivity.”
 Khanal, Samir. “Function point vs. Story point.” See
http://c2.com/cgi/wiki?NebulousUnitOfTime for the definition.
(Mike Cohn’s post)
(Giora Morein’s response); http://radio.javaranch.com/lasse/2008/04/17/1208381586654.html
 Cohn, Mike. Agile Estimating & Planning.