Exposure is how much light you let into the camera; it is controlled by the shutter speed (how long the shutter is open), the aperture (how large the opening is), and the ISO, which is the film (on a traditional camera) or sensor (on a digital camera)'s sensitivity to light.
With ISO, the higher the number, the more risk there is of grain in your photos, so as a general rule, you will want to try to keep the ISO as low as conditions will allow (unless, of course, you want a grainy photo for artistic or other reasons).
Actually, here is an article that explains the basics of exposure in photography.
ISO sensitivitiy is basically how quickly the sensor will "absorb" the captured light from the lens. If it's set high then light will be gathered quickly giving you lesser quality but faster shutter speed. So when you would need a higher ISO is basically when you need to shoot fast to avoid camera shake or subject movement, and you don't have the light to allow that without compromising image quality, ie hand-held or action shots in low light. If you can though, try to achieve your faster shutter speed with either more light (ie, strobes or speedlights) or a wider aperture (faster lens) before bumping up the ISO as a last resort.