When I was first learning to quilt, just a few years ago, I found this great visual under the quilting section of about.com that showed how fabrics are cut for quilting. It instantly helped me understand the concept. I've pinned this picture on Pinterest, and it gets re-pinned often, so I thought I would share it here as well.
How a Fat Quarter compares to a Regular Quarter cut of fabric.
Basically, a yard of fabric is cut at a length of 36" off of the bolt, which is typically around 42" to 44" across the width.
A quarter yard of fabric therefore, would be cut at a length of 9" (1/4 of a yard) across the full width (42" to 44"). This is the shown as the long blue rectangle in the above diagram. This type of cut is good when long strips of fabric are needed.
A Fat Quarter of fabric is still a fourth of the yard, but is cut at twice the length (18") by half the width (about 22"), so it is a "fatter' piece of fabric. You are able to cut larger pieces of fabric out of a Fat Quarter. It gives you more options.
The term Fat Quarter is one of the first concepts that is taught in quilting, so I'm sure this is nothing new to many of you. I just like this diagram and think it is a clear representation to show those that may not yet understand.