As I've said before, your camera will preform best at its lowest ISO. Therefore, it is best to set your camera to the lowest ISO setting you can under the conditions you are photographing. However, there are times when it is valuable, or even essential, to increase your ISO in order to capture a descent image. Consider this situation . . .
I took this picture at ISO 200 with a shutter speed of 1/2.5 and aperture of f/5.
If you look closely you can see that the picture is blurry because the shutter speed is too slow (see How to choose a Shutter Speed for more on this.)
If I kept all the other settings the same and increased my shutter speed, it would produce a darker image, which is not what I wanted. Another option (in theory) would be to increase the shutter speed (letting less light in) and make the aperture wider (letting more light in) to produce the same exposure with a faster shutter speed. Unfortunately, in this situation, the aperture was already as wide as it could go. Another option is to increase the ISO, making the sensor more sensitive meaning that less light is required to produce an equally bright image. Doing this would allow for a faster shutter speed (to eliminate the blur) without making the picture darker.
So I took this picture at ISO 400, shutter speed 1/4 and aperture f/5.
Still Blurry. Still need a faster shutter speed. Now ISO 800, shutter speed 1/10 and aperture f/5.