Use exponentially when you want to say that something's increasing quickly by large amounts. Your friends and colleagues will be pleased to hear that your vocabulary is growing exponentially.
The root of exponentially is the French verb exponere, meaning “to put out.” Think of a factory that puts out so many products its creations seem to increase exponentially. The mathematical origin of this word has to do with multiplying a number by itself so think also of rabbits, who are known for multiplying exponentially. Now imagine if rabbits worked in a factory... (No animals were harmed during this explanation).