A mole is a way to count things that are far too small to count. If you have a dozen eggs, you know you have 12 of them. If you have a mole of eggs, you know you have 602,200,000,000,000,000,000,000 of them. That's a uselessly huge number for everyday situations, but since atoms and molecules are so small, it works out well in chemistry. A mole of eggs would be larger than the entire solar system, but a mole of water molecules is around 18 grams of it, only slightly larger than a tablespoon.
Moles are used in chemistry because chemistry depends on having the right amounts of things that are hard to directly compare any other way. Say you want to turn oxygen (O) and hydrogen (H) into water (H2O). You know there are two atoms of hydrogen for every atom of oxygen in a water molecule. Say you assemble proportional masses of hydrogen and oxygen, something like 10 grams of hydrogen and 5 grams of oxygen. Once you react them, you'll find you only have 5.6 grams of water, not the 15 you'd expect. You'd also be left with 9.4 grams of hydrogen, completely unreacted. The reason for this is simple, oxygen is a lot bigger than hydrogen.
In that example, you've got 10 grams of hydrogen. The molar mass of hydrogen is 1.008 grams per mole, so you have 9.92 moles of hydrogen. The molar mass of oxygen on the other hand is 15.999 grams per mole, so you've only brought 0.31 moles of oxygen. When you make water, you need two dozen hydrogen atoms for every one dozen oxygen atoms to make one dozen water molecules. So, instead of having 20 dozen hydrogen atoms and 10 dozen oxygen atoms, you brought a dozen oxygen atoms and 32 dozen hydrogen atoms. You'll still only get a dozen water molecules, and the leftover 30 dozen hydrogen won't have anything to do.
Using moles though, it becomes easy. If we instead use 2 moles of hydrogen (2.016 grams) and a mole of oxygen (15.999 grams) we'll end up with 1 mole of water (18.016 grams) and nothing left over.
When you see a chemical equation, like "C3H8 + 5 O2 -> 3 CO2 + 4 H2O" you can read it as either atoms/molecules or moles. Either: "1 molecule of propane + 5 molecules of oxygen* yields 3 molecules of carbon dioxide and 4 molecules of water" or, "1 mole of propane (44 grams) + 4 moles of oxygen (160 grams) yields 3 moles of carbon dioxide (132 grams) + 4 moles of water (72 grams)." Both are true, the second one is merely the first scaled up 6.022*1023 times.
*The oxygen in the air around you is actually oxygen molecules (O2), not oxygen atoms (O). Each oxygen molecule is two atoms of oxygen bonded together. The same is true of hydrogen, (H2) but it didn't matter in the first example. I could have just as easily written "+ 10 O" instead of "+ 5 O2."
