What would common houshold bleach do to hair?

Bleach comes in different forms. There is chlorine bleach (the common household bleach) and oxygen bleach (such as hydrogen peroxide). These bleaches break the double bonds of the molecules that are in the visible light spectrum (the reason we can see the color). If that bond is reduced or broken, then that visible light (colors) is not seen. All in all, what bleach does to your hair (or clothes, etc.) is breaks down compounds in it. It can break down your hair to the point where it will literally dissolves, or weaken bonds in your scalp to the point where it will simply fall out. I think this website explains quite well (and more in depth, probably more confusing unless you know a lot about chemistry) the effects of bleach and how it works: