The mole is an SI unit of measure that denotes the number of atoms in 12 grams of carbon. Note that the molecular weight of carbon is 12 grams. Now we can understand that each element has one mole worth of atoms when you have the equivalent of its molecular weight. For instance, there is one mole of atoms in 16 grams of oxygen (whose molecular weight is 16). There is a small problem here--chemists assumed carbon had six neutrons and six protons, so 12 grams is a logical way of tying a system of weights to actual molecules. However, isotopes of carbon make a real one-mole sample of carbon slightly heavier, about 12.01 grams per mole. The same is true of all the elements on the periodic table--don't worry about this too much, and when figuring molarity, just use the values given for each element in the periodic table.
Molecules can be counted by moles just like atoms. For example, in sodium chloride, a mole would consist of the atomic weight of sodium plus the atomic weight of chloride. Added together, there is one mole of molecules in 58.43 grams of sodium chloride. The combined weight of the salt is known as the molecular weight. This is how much one mole of the molecule weighs.
Molarity is the number of moles of a substance dissolved in exactly one liter of solution. To make a one molar solution of sodium chloride, you would dissolve 58.43 grams of sodium chloride in one liter of water. Note that if you add salt to one liter of water, the solution would be more than one liter. Correct for this by adding the salt to less than one liter of water then adding water to make a total volume of one liter, including the salt.
To make solutions of a certain molarity that are not exactly one liter, correct the amount of solute you add proportionally to the final volume. For instance, to make 1/2 liter of a one molar solution, add one-half mole of solute. Multiply the molecular weight of the solute by the number of liters or fraction of a liter that the final volume will be, and that is how much solute to add. For instance, for 32 milliliters of a one molar solution, add 32ml/1000ml per liter times 58.43 grams = 1.869 grams of sodium chloride in 32 milliliters of solution. This gives a one molar solution of sodium chloride in 32ml of solution.
For solutions that are not one molar, correct the amount of solute added to a one liter solution proportionally. For instance, for a two-molar solution, add 2 x 58.43 grams of sodium chloride to a solution of one liter total volume. For a 1/2 molar solution, add 1/2 x 58.43 grams to one liter of solution.
For complex solutions, like a 32.00ml volume of 0.2800 molar sodium chloride, just multiply through all the proportions to know how much sodium chloride to add. 32.00ml / 1000ml per liter x 0.2800 moles x 58.43 grams per mole = 0.5235 grams of sodium chloride in 32ml of solution for a 0.28 molar solution.
Use the molecular weight for any molecule to determine how much solute to add for a desired molarity. Some chemicals, like sodium tetraborate, are not made up of just sodium and tetraborate. They have water molecules incorporated in their crystals. The container will list a formula weight, which is the molecular weight of the chemical plus the added waters. Use the formula weight to figure out how much to add for the proper molarity, not the theoretical molecular weight. For practice, what is the molarity of water? One liter of water weighs 1,000 grams. The molecular weight of water is 18.01 grams, so 1,000g per liter / 18.01 grams per mole = 55.52 moles per liter.