Storyboarding comes from movie making to describe the scenes of a film before the actual shots. It's essential in describing the general framework of the production: casting, movement, people, transitionts etc.
On the web and for multi-media projects, this method also has many advantages. In certain cases, wireframes - used to illustrate the general page layout and navigation of a site - are not enough to illustrate special areas whith a high amount of animation or interaction.
The story board allows the client to visualize and "feel" how the future site or application will eventually turn out. It can easily be modified and different scenarios can be explored, without a line of code. This of course helps to save precious time and money, by avoiding rework later and also providing a precise scenario to the developers / animators who will implement the concept.
As an example, here is a part of a storyboard I did recently for a client.