The Mole and Atomic Masses
Remembering that all atomic masses are relative numbers (carbon is 12 and hydrogen is 1 because carbon is 12 times as heavy as hydogen), we can make some simple assumptions about groups of atoms.
If one carbon is 12 times heavier than 1 hydrogen, then 10 carbons should be 12 times heavier than 10 hydrogens. Similarly a dozen carbons should weigh 12 times as muh as a dozen hydrogens and 100 carbons should weigh 12 times as much as 100 hydrogens.
Working the other direction, if I have some carbon and some hydrogen and if the carbon weighs 12 times as much at the hydrogen, I should have the same number of atoms of the two (maybe a dozen of each, maybe 100 of each, etc.) In this way, we can use the relative masses of the elements to figure out whether we have the same number of particles. So, 10 g of hydrogen should have the same number of atoms as 120 grams of carbon (12 times as heavy) and 30 lbs of hydrogen should have the same number of particles as 360 lbs of carbon (again, 12 times as heavy).
NOTE: we still don't have any idea how many particles there are--just that the number is the same.
Based on this idea, chemists created the mole. The idea is really quite simple. We already had a table filled with relative weights. We simply took those numbers (for example 12.011 for carbon) and put the unit grams after it. So a mole of carbon became 12.011 g of carbon, and a mole of hydrogen is 1.00794 g of hydrogen. (Notice that if you have a mole of carbon and a mole of hydrogen, the carbon will weigh 12 times as much). Each of these amounts (12.011 g of carbon or 1.00794 g of hydrogen) will contain the same number of particles--we didn't yet know how many, but we knew it was the same number.
This mass (the number from the periodic table in grams) was named the molar mass. So the molar mass of carbon is 12.011 g/mol and the molar mass of hydrogen is 1.00794 g/mol.
NOTE: the abbreviation for mole is mol. Yes, that is kind of pathetic, but that's what it is. m was already taken (meters and milli), ml was milliliters, so... we were left with mol.
The definition of a mole then is:
A mole is the number of particles present in a sample of a substance equal to its molar mass.
Of course, molar mass is not just confined to atoms and elements. It applies to molecules as well.