Profoundness is the characteristic of being full of meaningful insight. The profoundness of your favorite movie's final moments may make you cry every single time you watch it.
Profoundness and profound come from the Latin root profundus, which means "deep, vast," and also "obscure." This sense of obscurity, or being hard to understand, is evident in philosophical profoundness — you know it's deep and important, but you're not sure you totally get it. Another kind of profoundness is intensity. After you wake from a terrible nightmare, the profoundness of your fear may keep you frozen in terror for several minutes.