If you’re profoundly moved by the last chapter of that novel you’re reading, then you’re extremely moved — you feel it deep in your soul.
The adverb profoundly means something similar to “extremely,” with the additional sense that it’s something intense and deeply felt. If you’re profoundly confused, you’re very confused — confused in a way that seems bottomless. The word can also describe something that affects you greatly. If you’re profoundly influenced by the works of Shakespeare, that means his writing shaped the whole way you think about writing, whether you're working on plays, limericks, love letters.