An amalgam of the answers I liked best came from Umut Gokcen and Mario G. It went something like this, modified a little by me.

In physics a law of motion tell you exactly where a particle whose position you know right now will be at a very small instant of time later. Used again, the same law of motion then tells you where it will move to another instant later.

Calculus is the branch of mathematics that tells you how to add up all the small future exact position changes over instants of time to determine exactly where the particle will be in the future.

But not everything obeys such exact laws of motion. A stock price which you know today will not take some definite value an instant later, but rather will have a probability of being in some range of values. This price motion is called random or ‘stochastic.’

Stochastic calculus is an extension of calculus that tells you how to add up all the small future ranges of movements over each instant of time to determine the final range of values and probabilities the stock price can take at some future time.