ISO is one of the most important camera settings, you will significantly improve not just the quality of your photos, but also the variety of them too by making good use of the ISO setting.

Years ago, in the days of film, now being lost in the midst of time, the film manufacturers used to print a number on the side of the box. This number represented the speed of the film, that is, how sensitive the film was to light, and the higher the number, the more sensitive was the film to light. Using a higher ASA film made it much easier to take photos in low light or to take fast moving subjects, i.e sports or birds in flight. The problem was, especially in the early days of film, the higher the ISO, the more 'grain' was introduced to the negative and consequently the prints.

In this digital age, the term ASA has been replaced with ISO, and 'grain' is now referred to as 'noise'. With film, yo were locked into the ASA for the whole roll of film, the only way of varying the ASA while shooting was to carry another camera body loaded with a different film. How times have changed, we are now able to vary the ISO on a frame by frame basis, plus in my opinion the quality of high ISO images is now better than it was with film.

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Here's a simple analogy showing how the sensitivity of the camera sensor changes when you adjust the ISO. If you arrive at a cinema show a bit late, everything looks really dark when you go in doesn't it? But then this happens after a minute or two:

What exactly is the benefit of varying the ISO
Many people I've spoken to think that changing the ISO will lighten or darken the image, and although this is true when using the Manual mode, it's not true when you are using any of the semi-auto modes like Aperture / Shutter speed priority. That's because when you change the ISO, the camera will automatically compensate by changing one of the other settings. So instead, think of increasing the ISO as allowing you to take a photo that you wouldn't normally be able to take in the Auto mode (or at least, not without using a flash).

Low ISO values - 100, 200, 400 - Great quality
High ISO values - 800, 1600, 3200+ - More noise
Here's a rough idea of how upping the ISO increases the amount of noise.

So, generally speaking you should keep the ISO as low as possible for the best quality, the problem is that if you leave the ISO as a smaller number in low light, the chances are your shutter speed will drop to a slow speed, introducing camera shake or subject movement. In that case, raise the ISO as appropriate, and then (assuming your aperture is already at its maximum, i.e low number), your camera will compensate by switching to a faster shutter speed.

Very (very) rough guide to ISO values:-
ISO 400 - dusk
ISO 800 - indoors during daylight or shooting a fast moving subject
ISO 1600 - Well lit interior or shooting a very fast moving subject
ISO 3200+ - Church interiors or inside a building with not much natural light or night scene without a tripod