The Gillespie algorithm is a way of figuring out what will happen when a group of particles interact with each other. It works by putting the particles in a computer and having them move around randomly. As they move, they can occasionally bump into each other, and when they do, the computer makes a note of it. By repeating this process over and over again, the computer can use the notes it makes to figure out what kind of patterns the particles make, or predict what might happen in the future.