Does H2O2 break down into H2O + O or H2O + O2?
$\ce{H2O2}$ -> $\ce{H2O +1/2O2}$

This might make it easier to understand. 

$\ce{H2O2}$ will homolytically cleave for form two $\ce{.OH}$ radicals. Radicals are very reactive and will start a chain reaction, but ultimately you will end up with water and oxygen products from $\ce{H2O2}$. 

(Why does it form two hydroxy radicals? because the O-O single bond is weak and unstable - see MO theory)

How do we know this? Look at a bottle of $\ce{H2O2}$ that has been left out of the fridge for too long. There will be a slight pressure build up of $\ce{O2}$ gas and the % concentration $\ce{H2O2}$ will be down due to formation of $\ce{H2O}$and loss of $\ce{H2O2}$. 



For sake of completion, 
You can alternatively balance the equation thus:
$\ce{2H2O2 -&gt; 2H2O + O2}$