Since atomic number is the number of protons in an atom and atomic mass is the mass of protons, neutrons, and electrons in an atom, it seems intuitively obvious that increasing the number of protons would increase the atomic mass. However, if you look at the atomic masses on a periodic table you will see that cobalt (atomic number 27) is more massive than nickel (atomic number 28). Uranium (number 92) is more massive than neptunium (number 93). Different periodic tables even list different numbers for atomic masses. What's up with that, anyway?
The reason increasing atomic number doesn't always equate to increasing mass is because many atoms don't have a number of neutrons equal to the number of protons. In other words, several isotopes of an element may exist.
If a sizeable portion of an element of lower atomic number exists in the form of heavy isotopes, then the mass of that element may (overall) be heavier than that of the next element. If there were no isotopes and all elements had a number of neutrons equal to the number of protons, then atomic mass would be approximately twice the atomic number (approximately because protons and neutrons don't have exactly the same mass... the mass of electrons is so small that it is negligible). Different periodic tables give differing atomic masses because the percentages of isotopes of an element may be considered changed from one publication to another.
A molecule is a combination of two or more atoms that are held together by covalent bonds. A molecule is the smallest unit of a compound that still displays the properties associated with that compound. Molecules may contain two atoms of the same element, such as O2 and H2, or they may consist of two or more different atoms, such as CCl4 and H2O. In the study of chemistry, molecules are usually discussed in terms of their molecular weights and moles.
Ionic compounds, such as NaCl and KBr, do not form true molecules. In their solid state, these substances form a three-dimensional array of charged particles. In such a case, molecular weight has no meaning, so the term formula weight is used instead.
The molecular weight of a molecule is calculated by adding the atomic weights (in atomic mass units or amu) of the atoms in the molecule. The formula weight of an ionic compound is calculated by adding its atomic weights according to its empirical formula.
A mole is defined as the quantity of a substance that has the same number of particles as are found in 12.000 grams of carbon-12. This number, Avogadro's number, is 6.022x1023. The mass in grams of one mole of a compound is equal to the molecular weight of the compound in atomic mass units. One mole of a compound contains 6.022x1023 molecules of the compound. The mass of 1 mole of a compound is called its molar weight or molar mass. The units for molar weight or molar mass are grams per mole. Here is the formula to determing the number of moles of a sample
mol = weight of sample (g) / molar weight (g/mol)
A mole is simply a unit of measurement. Units are invented when existing units are inadequate. Chemical reactions often take place at levels where using grams wouldn't make sense, yet using absolute numbers of atoms/molecules/ions would be confusing, too. Like all units, a mole has to be based on something reproducible. A mole is the quantity of anything that has the same number of particles found in 12.000 grams of carbon-12. That number of particles is Avogadro's Number, which is roughly 6.02x1023. A mole of carbon atoms is 6.02x1023 carbon atoms. A mole of chemistry teachers is 6.02x1023 chemistry teachers. It's a lot easier to write the word 'mole' than to write '6.02x1023' anytime you want to refer to a large number of things! Basically, that's why this particular unit was invented.
Why don't we simply stick with units like grams (and nanograms and kilograms, etc.)? The answer is that moles give us a consistent method to convert between atoms/molecules and grams. It's simply a convenient unit to use when performing calculations. Okay... you may not find it too convenient when you are first learning how to use it, but once you become familiar with it, a mole will be as normal a unit as, say, a dozen or a byte.