ISO is the measurement of how sensitive your camera's sensor is to light.

The larger the ISO number, the more sensitive. So ISO 1600 is wayyy more sensitive to light than ISO 200.

The more sensitive the ISO, the more light is collected by the sensor, and thus the brighter the image.

When do I use it?
If you've already adjusted your shutter speed and aperture to exactly what you want, and your picture is still underexposed, time to adjust your ISO.

The reason I would advise this as the last resort when adjusting your camera settings is because the more sensitive the camera sensor is, the grainier your pictures can be.

So, the higher your ISO number, the grainier your pictures will tend to get.

Here are photos I took of my lovely Christmas tree, with a gradually increasing ISO. I started at ISO 100, and then kept going up from there. 

Not to be confused... to keep the picture well exposed (instead of getting brighter and brighter) I was adjusting my shutter speed and my other settings were constant.
Lens: 85mm F/1.8
Shutter Speed at ISO 100: 1/30
Shutter Speed at ISO 200: 1/30 (To be fair, this one ended up overexposed and I had to edit it)

Shutter Speed at ISO 400: 1/60

Shutter Speed at ISO 800: 1/100

Shutter Speed at ISO 1600:1/200

Shutter Speed at ISO 3200: 1/400
Shutter Speed at ISO 6400: 1/1600

What I want you to really get out of this though, it to watch how the pictures got grainier as my ISO increased:

Note: Different cameras, and different lighting situations will effect how grainier a picture gets at certain ISOs. If you're starting with a beginner DSLR, your ISO won't be able to go up near as high without getting very grainy looking.

What to take away from this?
The higher the ISO, the more light in the picture, and the grainier it may get.
If you need more light though, then this is a sacrifice you can make 

EXERCISE: (Because you need to go out and do it yourself)

Sit down in your living room at night, and set your shutter speed to 1/80, and your aperture to the lowest your lens will allow. Begin at ISO 100, and go up incrementally from there. Don't adjust any of your other settings, and see what happens to your image.

After you've done that, incrementally adjust your ISO back down, and adjust your aperture and shutter speed accordingly to try and have a well exposed picture.

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