Khagne is the second year of the preparation for the competitive exam to enter the Ecole Normale Superieure (ENS). The first year is called Hypokhagne. It does not permit you to do anything, unless you pass the competitive exam to enter ENS, in which case, you still have a couple more years to do, at least, but with an excellent preparation that permits you to pass the following competitive exams, especially the agregation, with flying colors, and which in the end lead to teaching. Then you are "normalienne", like Simone de Beauvoir and other prominent intellectuals, and it opens all kinds of doors to you. In other words, it means you're really smart, but means also you had the discipline to endure intensive prepping and kind of postpone adulthood (in terms of the freedom of the university life as opposed to the grueling schedule of classes in the Hypo and Khagne) till you got in there.
After Hypokhagne and Khagne, you have one more year to acquire your "licence", which you can do in a university if you have failed the exam to the ENS. The licence is the equivalent of the Bachelor of Arts degree which in the U.S takes 4 years to acquire. You will still need to acquire your master's degree and pass some competitive exams such as the CAPES or the Agregation, otherwise you are doomed to a life of substituting. And of course, you can go on to a doctorate.
You usually talk about the "two years of Khagne". To prep the students more efficiently, they are done in a reputable high school, Fenelon being one of the finest. The classes are small, you get a lot of attention, it's both very competitive and clannish at the same time, and in my day (I was at Fenelon too), they didn't hesitate to use a little humiliation here and there in front of your peers to encourage you to do better.
Diplomatic studies are done through the ENA (Ecole Nationale d'Administration) which also requires a competitive exam to get in, a very hard one that emphasizes general culture and economics, finance, management, history, current events. You usually get in after passing the first year of Sciences Politiques (Sciences Po), which Caroline failed when she attempted it. It's a lot of memorization and non stop studying. But then you can end up President of France (not that Charlotte would want to...)
I'm just trying to answer CaliforniaDreamin's question. I know it does not belong on this thread. Feel free to move it !
Just realized : didn't mean to brag with the business of being very smart etc. I failed the competitive exam at the end of the 2 years, just like Charlotte ! Entered the third year of licence at the Sorbonne after that. No need to feel ashamed (nor superior !)