The ISO setting adjusts how sensitive the camera is to light. The higher the setting, the more sensitive the camera is - with higher settings you can use faster shutter speeds, or take pictures when there is less light, for example. They are light the film speeds that you used to get on cameras, only you can vary the setting from shot-to-shot.
As you increase the ISO setting, an amplifier in your camera boosts the signal coming from the sensor. If the signal is boosted too much, you start to get noise - this looks like an analogue television with a weak signal, or a grainy picture taken on film. For the highest quality, therefore, it's usually best to leave your camera on the lowest setting.
You would increase the ISO setting where there isn't enough light for you to be able to take the picture you want. For example, if you want to take a picture indoors and the camera is selecting a shutter speed that is too slow for you to be able hold the camera still, or too slow to freeze any movement. Or, if you're at a sporting event and you want both a large depth of field (i.e. you need an aperture with a large f number) and a fast shutter speed, but there isn't enough light for you to select both and get the correct exposure.