The ISO number is a measure of light sensitivity. It originally referred to the sensitivity of a given type of film, and the standards for measuring were determined by the International Standards Organization (ISO), which is where the name comes from. In a film camera, you had to change film to change ISO. Digital cameras allow you to change ISO through the camera's menu functions, adjusting the sensitivity of the camera's sensor to an ISO equivalent number.
So how does ISO work? While it measures light sensitivity, photographers refer to ISO as the "speed" of the film or sensor. At high sensitivity, more light is sensed within a given period of time than at low sensitivity, so high sensitivity is considered faster. Unlike aperture, ISO settings are relatively straightforward. Low ISO numbers indicate the least amount of light sensitivity, while high ISO numbers are faster, more sensitive settings.
Why not always use the highest ISO possible all the time? In film cameras, high ISO film was grainy. We didn't escape that limitation with digital cameras, but instead of grain, high ISO numbers introduce digital noise. One of the most important things a photographer can learn is how to get the best quality shot in a given lighting condition with the lowest possible ISO setting. Of course, sporting events and other fast moving action requires high, fast ISO numbers. Fortunately, those situations are usually brightly lit.