Mood boards, very roughly, are to design what high-level block diagrams are to programming: a way to pull the basic elements of together so you can have a broad agreement on what the end result will look like.
There was a great example of putting together a mood board in the Day 1 keynote at Adobe MAX this year, when Mark Magner from Sesame Workshop demonstrated using a couple of the new Adobe touch apps. His segment starts at about 15:30 into the video. What's instructive here is not only the use of a mood board, but also the end product that resulted from it.
Mood boards are essential when you have a large project that will take a lot of work to develop, and you need to be sure that you and the client are on the same page before you invest tons of time. They're also great for just playing with ideas while searching for that "Ah-ha!" moment when you suddenly know what direction you need to go with on a design. It gets you past that initial period with a new project where you feel completely blank, can't get a handle on it, and wonder if your entire career up to that point has been a fluke and maybe you weren't really cut out to be a designer after all. (It happens to all of us.)