Okay, in simplest English, the manganese dioxide works catalytically (without undergoing permanent change itself) to cause the hydrogen peroxide to break down into water and monatomic oxygen radicals (MORs). The monatomic oxygen then unites to form normal diatomic oxygen molecules. Because you need two MORs to make one oxygen molecule, you need to write the values of the equation so that you have enough oxygen to produce the ultimate result: either saying (my first equation) that the first step of the process is to produce the MORs, or, preferentially, saying (my second equation) that two moles of peroxide produces two moles of water and one mole of oxygen.
A mole is a quantity of a substance (usually measured in grams) that is equivalent to the molecular weight of the substance in the unit of measure. In other words, it’s how much of that substance contains Avogadro’s number of molecules of that substance (when talking grams), or a multiple of it (when talking another unit of measure).
Industrial equations might substitute tons for grams; descriptions of chemical cellular processes might be talking about micrograms in each cell. But in any chemical process, one mole is equivalent to the amount of the substance it takes to make up one “unit” of that substance from the reaction POV.
Like this: the elementary school class goes to the ranch, and the kids are treated to horseback rides. They need to have as many horses available as there are kids. Obviously, 30 horses weigh much more than 30 third graders – but they’re in one-to-one correspondence.
To get a chemical equation to balance, you need to put as many of each component on one side as you have on the other. This may mean that you need twice, three times, or one-and-a-half times as much of a given starting substance as you have of one of the products, as we had here: it takes two moles of peroxide to produce two moles of water plus one mole of normal oxygen.
You need equal numbers of each kind of atom on each side of the equation. If one product is something that doesn’t match the quantities (as for example the O[sub]2[/sub] in the equation), you need to alter the quantities to produce a result measurable in whole units of that product. (I suppose you could write the equation as
H[sub]2[/sub]O[sub]2[/sub] —> H[sub]2[/sub]O + 0.5O[sub]2[/sub]
but the custom is not to work with such fractional units.)