Remember film speed? Well, ISO controls how sensitive your sensor is to the light. This page has a discussion that describes how it works: http://forums.cnet.com/7723-7593_102-363839.html

Auto ISO? Used to be frowned on, and rightly so...but things are changing because the sensors are improving every version. There are also controls that limit how hight the auto setting will go.

The ISO setting is seductive. The higher the number the more sensitive your sensor becomes so you can take photos in darker conditions. However, there is a trade-off... generally higher ISO settings increase the "noise" in the image. Digital noise is dots of light and color in the image that degrade the look, especially in the shadow areas. If it is too extreme it can ruin the image. However, the newest cameras have extrodinary performance at high ISO settings. Do test shots with your camera to know where you draw the line on noise.

Quality? The highest quality setting is the lowest numerical setting for your camera that has a number...In the same way that higher speed films had more visible grain, higher ISOs can create more noise (speckles of light and color often in the shadow regions).

Noise ruins this exposure from Canon 20D at 1600ISO.

Noise can be detected in this exposure from Nikon D3 at 4000 ISO.

ISO is part one of the the exposure triangle. (ISO - Shutter Speed - Aperature Size)

Full stops, 1/3 stops, 1/2 stops? What is a stop? Explained in the Exposure section. But know now that your camera can probably be set to use these settings for ISO.

My Digital Photography School article on learning to set your ISO in almost any condition: http://digital-photography-school.com/use-program-mode-set-iso/

When shooting:

What is the best choice for quality?

You need to experiment with your camera before setting AUTO ISO parameters. Then you will know what limit to put on the AUTO ISO setting.

Are you a "pixel peeper?" Rick Sammon's quote on noise in a photo is worth noting.

What would be your normal?