Question:   What does the law of multiple proportions mean?

Answer:

Let's see an example:

Hydrogen can react with oxygen in a couple of different ways.  One way occurs when you take two moles of hydrogen and react it with one mole of oxygen.  When you do this, you make H₂O.  Another way occurs when you take one mole of hydrogen and react it with one mole of oxygen:  When you do this, you make H₂O₂.

In the first case, two moles of hydrogen weigh 4 grams and one mole of oxygen weighs 32 grams.  The ratio of
oxygen to hydrogen is 8:1. In the second case, one mole of hydrogen weighs 2 grams and one mole of oxygen weighs 32 grams.  The ratio of oxygen to hydrogen is 16:1.  Since 16 is equal to 8 x 2, this satisfies the law of multiple proportions.

What you'll find is that any time two elements can form more than one compound with each other, this kind of relationship between the masses takes place.