Brevet is an exam you pass in French at the end of "collège" (middle school) which is 'troisième' in French. The final grade is composed of your French, math and history grade plus an oral you have to take in front of a jury. The brevet is mainly considered useless as it is very easy and unless you fail your grade won't matter at all in your future studies.

Personally I think it's just to get used to exams and doesn't really have a purpose. You may have to redo your final year of middle school if you fail it but that rarely happens or they could keep you from entering a general high school and force you to go in to a technology or professional hs