You have to understand all the basic concepts of jiujitsu and be able to apply them in different situations.
White belt = dont understand that there are governing concepts in bjj. Learning is technique based.
Blue belt = Understand that concepts >technique; starting to learn different concepts as applied to limited situations. Can start to apply concepts to situations you have been in instead of relying on techniques.
Purple belt = Understand major concepts; can apply those major concepts in a variety of situations. Rely on conceptual understanding to learn bjj rather than technical learnings.
Brown belt = Have a general, broad understanding of all bjj underlying concepts; Can apply said concepts to know what to do in entirely new situations.
Black belt = Understand in detail all bjj concepts, can apply in any situation. (obviously this takes years and years of refinement; like saulo says black belt is your longest belt).
Thats my opinion - thats why they say black belt is when you start learning jiujitsu. You finally understand all the concepts in detail, and all that is left is training and practicing and applying in different ways to expand and sharpen your knowledge and skills.
Its not very eloquent, but the best example I can think of is by considering a seminar. A white belt will learn some techniques. a Blue belt will start seeing some completely new positions and ideas which change the way they think. A purple belt will use the techniques demonstrated to extract concepts which can shape their game further. A brown belt will extract concepts in order to refine their understanding and apply that to their own game. A black belt will do the same, except on a higher level. Basically purple and up is not about learning techniques (although of course you do). Its about learning how to apply the same idea to different situations.
If you can do that, your growth is exponential, while technique based growth is simply linear.
Final example: white = high school; blue = college; purple = masters program; brown = doctorate program.
So getting your black belt is like getting a phd in bjj. By the time you do post graduate studies, you are dealing with conceptual theories and applications.