Notwithstanding means "in spite of something." Your boasts about having memorized the entire textbook and bribing the teacher with apples notwithstanding, you still managed to fail the final exam with flying colors.
You don't have to look too closely at the word to be able to break it into three parts: not- + withstand (to successfully oppose or resist) + the participial ending -ing. In the most literal sense, notwithstanding actually means to successfully oppose or resist. Top-of-the-line wet-traction tires notwithstanding, the car still got stuck in the mud. Notwithstanding works just as well in a more figurative context. His reputation for charm and tact notwithstanding, he offended everyone in the room.