Molar mass is how much mass one mole of a given compound/element has. While a mole is a fixed unit of measure, the molar mass will vary from element to element.
Following Mojo's example of a mole being a similar unit of measure to a dozen (in that a dozen = 12, and a mole = 6.02 x 10^23, or Avogadro's Number):
Let's say you have three dozen eggs. One dozen small, one dozen medium, and one dozen large eggs.
A dozen small eggs weigh 16 ounces.
A dozen medium eggs weigh 20 ounces.
A dozen large eggs weigh 24 ounces.
In all three of these, you still have the same quantity of eggs -- 12 of each size. However, each dozen eggs has a different mass because the eggs themselves are different. Such is the same with elements; one atom of carbon has a different mass than one atom of hydrogen. Thus, one mole of carbon has a different mass than one mole of hydrogen. Like Jay said, moles are simply a much more convenient unit of measure to use when dealing with elements since atoms are so ridiculously small -- moles simply let you "scale up" your numbers.
Number of moles = (# of grams) ÷ (molar mass)
Number of grams = (# of moles) × (molar mass)
This is where math comes into play. Moles, molar mass, and the resulting mass of a sample is a balanced equation.
Here's an example. Carbon has a molar mass of 12.0107 grams per mole. That is, 6.02 x 10^23 atoms -- or one mole -- of carbon has a mass of 12.0107 grams. Now, let's say you have 3 moles of carbon. How many grams of it do you have? Since we know the mass of one mole (12.0107 grams), simply multiply it by 3 (the number of moles you have), and you'll end up with 36.0321 grams.
Here's another example using the egg analogy. Let's say a dozen medium eggs have a mass of 20 ounces. You have 3 dozen eggs; how much mass do all of them have? 20 ounces per dozen, multiplied by 3 dozen, equals 60 ounces.
Now, let's say you have 48 ounces of large eggs. As said above, large eggs have a mass of 24 ounces per dozen. How many dozen eggs do you have? Two dozen -- 48 ounces (how much you have total) divided by 24 ounces (how much in a dozen) is two.
Atomic mass (remember, there's difference between mass and weight) is the mass of a single atom of an element. The unit of measure is different, it's called an AMU, or Atomic Mass Unit. An AMU is a *very* small number; if you tried to convert it into grams you'd have a decimal point with a shit ton of zeros to the right of it. For the most part, you'll be working with molar mass, since it's such a much easier unit of measure to work with. My wife is a chemist, and even she doesn't use AMU's very often, and I suspect Jay will say the same.
