A mol is the number of atoms or molecules it takes to make up the the atomic mass (just think "weight" for now) or molecular mass/weight in grams.  E.g., since the atomic weight of oxygen is 16, the "gram atomic weight" of oxygen is 16 grams.  It will "turn out" that it is the same number of atoms needed to make up 16 grams of oxygen as make up 12 grams of carbon or 1 gram of hydrogen, or "N" grams of any element with molecular weight "N". 

Rember a mol is a number-- the number of atoms or molecules or some such that make up a certain weight.  Don't confuse the different ways of measuring "amounts" or "quantities" in chemistry: volumes, masses (weights), number of "particles", etc.  A mol is a number of particles, but it is tied to weight in a screwy way because it is the number of particles required to give a certain weight, but the weight is not a constant number.  The weight is relative to the atomic weights (or molecular weights) of the different elements and molecules.  So if you have found it confusing, that is because it is; it is a weird way of giving a quantity-weight ratio.

For the first time, I figured out this morning why this "mol stuff" is important and why it works. I never saw before why it was the same number of atoms or molecules that gave each element or each molecule its atomic mass, molecular mass, or its "formula mass" in grams (or any other amount or unit of mass or weight). Now I see it, finally.  Moles give you a way of determining the ratio of the number of atoms of an element (or molecules in a compound) to their weights. That is important when you go to mix stuff to make compounds, so you know you have the right "amount" of stuff -- which will involve how many atoms or molecules you are mixing; but which will be determined by you by weight, since you can't count the number of atoms or molecules directly.  To mix elements into compounds you need to go by how many atoms you are mixing together, not just how much weight you are mixing together of each element.  If you want to pair up marbles and bowling balls, you would need to have the same number of each.  But if you were unable to see them to count them, but could only see the weights of the boxes a bunch of bowling balls were in and the weights of the boxes a bunch of marbles were in, you could figure out a way to get the same number of bowling balls and marbles if you had some way of figuring how the weights of some number of bowling balls compared to the weights of some number of marbles.  That is all that is going on with the "mol stuff" -- it is a way of figuring out numerical quantities based on relative weights.

They could give you this ratio on the periodic chart in a different way, by just telling you how many atoms it took to make some mass, say one gram, but then it would be a different number of atoms for each element and each compound.  The weights would be the same but the number of "particles" would be different to get that weight.  That would be an intuitive way to do this and it would be in the form of atoms/gram.  But they didn't do that.
What they did was to say how many atoms or molecules it takes to equal its atomic mass or molecular mass in grams.  That way you get the same number of molecules but different masses.  This way is not as "normal" or intuitive but it is the same sort of thing.  At this time I don't see why the less intuitive way is better -- if it is.

1) I thought the following at first this morning, but I decided afterward (see 1A below, this may not quite be on target. Yet it may still be helpful to begin.) The reason why the same number of atoms of each element will give its atomic mass in grams (or any other unit of measure of mass) is that for atoms all you essentially are weighing are their protons and neutrons; and since these are supposedly the same things in all elements, and since they supposedly weigh the same as each other (i.e., the weight or mass of one proton = the weight or mass of any other proton or of any neutron, no matter what element it comes from) of course the weights/masses will work out the same. E.g., if Carbon has 12 total neutrons and protons together per atom and if Oxygen has 16 total per atom, then if we have the same number, of atoms of carbon as we do of oxygen, the total weight or mass will be in the proportion of 12 to 16, whether we have 1 atom of each, 10 atoms of each, or a million atoms of each. A million atoms of Carbon will be 12 million combined protons and neutrons, and a million atoms of oxygen will be 16 million combined protons and neutrons. So the ratio of their weights/masses will still be 12:16, whether this turns out to be grams, milligrams, kilograms, micro ounces or tons. 

Corollary revelation: Isn't it both interesting and weird that all the elements - which act so differently from each other - are made up of the same things, just in different numbers and arrangements. The reason that is odd is that if we have one bowling ball, or we have 100 bowling balls together, they are still just bowling balls. Now 100 tied together will not roll and will not be round, but they will still be bowling balls. But if you have one proton, you have hydrogen, whereas if you have 100 protons, you have a totally different element that feels, looks, and acts very very differently from hydrogen. That seems weird.

1A) Actually, however, the above is not quite the reason why the same number of atoms will make the gram-atomic-weight for each element.  It is just a specific case of the more general principle, where the individual elements that comprise the mass would not have to be the same.  The atomic masses or weights of any given atom will always be in some proportion to each other, even if those masses did not depend on the number of neutrons and/or protons, or even if different element protons or neutrons weighed different amounts.  Suppose we have elements A and B.  And suppose 1 atom of B weighs 27  times more than one atom of A, for whatever reason, not just because of protons and neutrons.  If we define a mole as the number of atoms that are required to make an element weigh its atomic weight/mass in grams, then however many atoms of A it takes to equal its atomic weight in grams, that same number of atoms of B will weigh 27 times more.  And therefore that number of atoms will make B weigh its atomic weight in grams.  Because atomic weights or masses of each atom of each element are proportional to each other, the same number of atoms of each element will give masses that are also proportional to each other.

2) The reason it works then for molecules - that is, the reason why the same number of molecules of each compound will give its molecular or formula mass in grams - is that molecules are just made up of atoms, and however many atoms you need to give the molecular weight of the compound in grams will be the same number of atoms you needed to have in their elemental form before you combined them.  E.g., if you have 20 molecules of, say sodium chloride, NaCl, you have twenty atoms of sodium and 20 atoms of chlorine combined, which will be 20 times the weight of one sodium atom and one chlorine atom, whether combined or not. And since one mole of chlorine contains the same number of atoms as one mole of sodium -- 6.024 x 10²³ atoms of each-- if you combine the two, you will have 6.024 x 10²³ molecules of sodium chloride, and the mass of that mole of sodium chloride will weigh the same as the combined mass of the individual sodium atoms and chlorine atoms. If we look at a molecule, such as water, H₂0, that does not have equal parts of different elements, it will come out this way: it takes twice as many hydrogen atoms as it does oxygen atoms to make water. If you start with 20 oxygen atoms, you will need 40 hydrogen atoms to make the water and you will get 20 molecules of water.  If you start with a mole of oxygen atoms (6.024 x 10²³) then you will need two moles of hydrogen atoms (2 x 6.024 x 10²³) and you will get one mole (6.024 x 10²³) of water molecules. Since you put in one gram atomic mass worth of oxygen atoms (16 grams) and two atomic gram masses worth of hydrogen atoms (2 grams), that will end up weighing 18 grams, which is the same as the gram molecular mass of that many molecules of water.  There is probably a way of generalizing this symbolically and mathematically, but I cannot see it right now.

3) The reason moles are important in chemistry is: suppose we want to make sodium chloride, or we want to make water, H₂O, we need to combine the right number of atoms of sodium and chlorine, or the right number of atoms of hydrogen and oxygen. But since it is difficult to count the atoms, it is easier to combine them by weights if we know how much weight will give the same number of atoms of each. That is what the concept of mol allows you to do, because we know that if we combine 2 pounds of Hydrogen with 16 pounds of oxygen, we will have twice as many hydrogen atoms as we have oxygen atoms and it will give us 18 pounds of water with no atoms of either element left over. (Well, maybe one or two or a thousand, if we didn't measure exactly right, of course, which is just about impossible to do.) See, you can't just put in twice the weight of hydrogen as you do oxygen, because that won't give you the right number of atoms, since it takes a lot more hydrogen atoms to make a pound then it takes oxygen atoms - just like it takes a lot more marbles to make 50 pounds then it takes bowling balls. Since bowling balls weigh a zillion times more than marbles, you need a zillion marbles to equal the weight of one bowling ball, and you therefore need a zillion times as many marbles to make 50 pounds than it takes bowling balls to make 50 pounds.

Now, if you know that one mole of any element gives the same number of atoms of that element as one mole of any other element does, and since one mole of any element equals its atomic weight in grams, you can then figure out how many grams or milligrams, or tons, of any elements you need in order to give you the right number of atoms of each to combine into molecules. That is why the mole stuff is important. It lets you calculate the relative number of atoms of elements from their relative weights or masses because it lets you know the ratios of the number of atoms or molecules to their  weight or mass.

PS, I am not sure if it is supposed to be "mols" or "moles", so sorry if I have the spelling wrong.

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