I am asking this question on behalf of my seven year old daughter, so if possible please take that into consideration in the answer!
I am not a chemist, but I do try to teach my daughter the basics of diverse subjects. I explained to her that the right-most elements on Mendeleev's chart do not generally react with other elements as they have all the electrons that they need. Those to the left of them need a single electron to be satisfied, those another column to the left need two electrons, and so on. Additionally, those elements on the left of the chart have an extra electron that they want to be rid of, and those one column to the right of them have two electrons to get rid of.
On this basis she understands why water has a single oxygen atom, but two hydrogen atoms. Then she asks my why hydrogen peroxide has only two hydrogen atoms. What satisfies the second oxygen atom? I don't know myself, I'm not a chemist!
Does hydrogen peroxide use a different type of bond? Is the electron-swapping somehow satisfied for hydrogen peroxide in a different way?
Googling around I discovered hydrogen bonding but this doesn't seem to work with Oxygen. What else should I be reading? I've googled and found many "Chemistry for Kids" websites, but they seem to be either (fun) experiments, or explanations that are no more simple than those in wikipedia and obviously not aimed at children.