In low light conditions, we often find our pictures are too dark. If you can't slow your shutter speed or decrease the number of your aperture then you have to increase your ISO.
ISO is like the film speeds of old. The higher the number, the more sensitive the camera becomes to available light. Remember the old 100, 200 and 400 ASA films? It works the same way but is using the digital sensor in the camera, not film.
The clever among you might start to think, "Okay, I'll just leave my camera set on ISO 1600 and I'll never have a darkness problem!" Well, that's true but you will probably have a brightness problem in the day and you will definitely suffer the nasy side effects of high ISO.
Can you see all the dots in this photo? That's digital noise. Just like the film grain in old cameras, high ISO results in digital noise in new ones.
As a result, the rule with ISO is to keep it as low as possible. Always try to use shutter speed and aperture to correct light where you can.
ISO is a method of last resort. I never call it your "get of jail for free" card because the cost of using it is that the higher your ISO, the more noise will appear - to the point where the photo is just not useable (like the one on the left).