a startup is an organization formed to search for a repeatable and scalable business model.
A good post by Steve Blank on what a Business Model is and other good stuff that any business person should be cognizant of.
How Does Customer Development, Agile Development and Lean Startups Fit?
The Customer Development process is the way startups quickly iterate and test each element of their business model. Agile Development is the way startups quickly iterate their product as they learn. A Lean Startup is Eric Ries’s description of the intersection of Customer Development, Agile Development and if available, open platforms and open source. (This methodology does for startups what the Toyota Lean Production System did for cars.)