ISO in digital photography is a measure of the sensitivity of the image sensor. ISO is measures in numbers like 100, 200, 400, 800, 1000, 1600 and higher. The higher the ISO number, the higher the sensitivity to light and more noise or static will appear in your image.
Higher sensitivity to light will allow you to use a faster shutter speed in darker conditions like indoors and at night. If your shutter speed was slower, your subject may be blurry or the image itself may be blurry from your shaky hands.
Outdoors on a sunny day, I typically set my ISO to 200 or 400. When I was taking pictures indoors at concerts and open mics without a tripod, I would usually have the ISO set to 800 or higher so that the shutter speed could be faster and I could get a picture of the performer that wasn’t blurry.
Here are examples of pictures with ISO the set at 200 and 3200. Click the pictures to see them bigger.

Clouds ISO 200

Clouds ISO 3200
Don’t be afraid to use a higher ISO (800 or so) with your digital camera. It’s often difficult to notice a difference unless the image is cropped a lot or it gets printed really large. Check out this post at the Digital Photography School website which explains how to choose the right ISO for your digital photography.

Bricks ISO 200

Bricks ISO 3200

Bricks Cropped ISO 200

Bricks Cropped ISO 3200
If you have any images with a high ISO that look wonderful, share a link in the comments!
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