Percentage Composition

The "mass of that element in one mole" is just the atomic mass multiplied by its subscript in the formula — the same quantity you compute along the way to finding molar mass. Once you've already found $M$ for a compound, percent composition is one more division for each element. Using one mole is a convenience, not a requirement: any sample size gives the same fraction, and one mole is the sample whose mass you already know ($M$ grams) and whose element masses you can read straight off the formula. In water, one mole ($18.02\text{ g}$) holds $2.02\text{ g}$ of hydrogen, so hydrogen is $\frac{2.02}{18.02}=11.21\%$ of the mass.

The law of definite proportions says that a pure compound always contains the same elements in the same proportions by mass, no matter where it came from or how much you have. Water from a glacier, a laboratory or a car's exhaust is always $11.21\%$ hydrogen and $88.79\%$ oxygen by mass. The reason is the fixed formula: every molecule has exactly two H atoms and one O atom, so doubling the sample doubles both masses and leaves the fraction unchanged. This is also why percent composition can identify a compound: two different compounds of the same elements (like CO and ${\text{CO}}_2$) have different formulas and therefore different percentages.

In the lab you usually don't know the formula yet; you measure masses. Then percent composition comes straight from the masses: $\%\text{element}=\frac{\text{mass of element}}{\text{mass of compound}}\times 100\%\text{.}$ For example, burning $0.486\text{ g}$ of magnesium ribbon produces $0.806\text{ g}$ of magnesium oxide, so the oxide is $\frac{0.486}{0.806}\times 100\%=60.3\%$ Mg. The formula MgO predicts $\frac{24.31}{40.31}\times 100\%=60.31\%$ Mg, so the experiment supports the formula MgO. A result that differs noticeably from the prediction points either to an experimental error or to a different formula. In this particular experiment, unburned magnesium and oxide smoke escaping from the crucible both make the product weigh too little, which pushes the calculated %Mg too high; a product that absorbed moisture would weigh too much and push %Mg too low.

Iron forms three common oxides, and percent composition tells them apart: FeO is $77.73\%$ Fe, ${\text{Fe}}_3{\text{O}}_4$ (magnetite) is $72.36\%$ Fe, and ${\text{Fe}}_2{\text{O}}_3$ (rust, hematite) is $69.94\%$ Fe. Mining companies assay ore samples in exactly this way to decide how much metal a deposit can yield. The fertilizer table above works the same way in reverse: a farmer who needs $50\text{ kg}$ of nitrogen on a field needs far less ammonia ($82.22\%$ N) than ammonium sulfate ($21.20\%$ N) to supply it. Percent composition is the number that lets products with different formulas be compared on a fair, per-kilogram basis, which is why it appears on fertilizer bags, food labels and ore reports.