Exponential describes a very rapid increase. An exponential rise in the number of first graders with head lice might be a result of all the hat sharing that's gone on lately.
Exponential is also a mathematical term, meaning "involving an exponent." When you raise a number to the tenth power, for example, that's an exponential increase in that number. When your next door neighbor describes the exponential growth of her rose bushes or the exponential increase in members of her book group, she means that there's been a sharp, fast expansion of both things.