A mole is defined as an Avogadro’s Number amount of…well, anything.
A’s# is the number of atoms of Carbon 12 in exactly 12 grams.
Can a similar or related number be found in exactly 16 grams of Oxygen 16?
yep
Yes indeed, and it is the exact same number, 6.022 x 10^23. That because each atom of O-16 weighs 16 atomic mass units (a “normal” hydrogen atom weighs roughly 1 a.m.u.) while C-12 weighs 12 a.m.u. Indeed, that is essentially the definition of Avogadro’s number: the number of particles needed for the mass in grams to be equal to the mass in a.m.u. of a single particle. So a mole of hydrogen weighs 1 g, a mole of gold weighs 197 g.
A mole of Hydrogen with one proton and no neutrons ( [sup]1[/sup]H ) would weight 1 g, and a mole of gold with 79 protons and 118 neutrons ( [sup]197[/sup]Au ) would weigh 197 g. However, in the real world, there are isotopes of both elements. That means that in any natural sample, the weight will be different; 1.00794 g/mol for hydrogen and 196.96655 g/mol for gold.
I only bring it up because Enola mentioned isotopes.
If you really want to get picky, the molar mass only comes out to an exact number of grams for [sup]12[/sup]C. This is because the atomic mass unit is defined to be 1/12 of the mass of a [sup]12[/sup]C atom. Other atoms have non-integral masses because of differences in the binding energy (the mass associated with the energy it would take to dismantle the atom) and the fact that neutrons and protons don’t have the same mass to start with.
Molecules also have a molar weight obtained by adding up the atomic masses of all the atoms involved. A mole of H[sub]2[/sub]O is 18 g.
Chava
Also, there will be some minor difference due to the fact that some of the energy of mass goes into the binding energy – i.e., the mass of a nucleus is always less than the sum of the masses of the individual protons and neutrons. Protons mass 1.00728u (and 1.00728 grams of them would be one mole (6.022x10^23) of protons). Neutrons mass 1.00866u. Two protons and two neutrons “should” mass 4.03188u, but due to the binding energy, an alpha particle masses 4.00153u. I’ve ignored the weights of electrons in this example (electrons mass about 1/2000 as much as a proton), but you get the idea.
But yes, the mass given in atomic mass units – which can usually be found on a periodic table as a number approximately double the atomic number with four or so decimals – tells how many grams would contain a mole of that element. The reason that carbon (more precisely, carbon12) is an unusually precise 12.0000 is that this was the substance used to define the atomic mass unit.
-b
On preview I see that Chava already covered much of this. Damn, I hate being slow of the mark! 
Chava: I want to get really picky.
You have correctly indentified the nuclear binding energy and its effect on atomic mass.
You have ignored chemical binding energy and its effect on molecular mass.
A mole is just a conversion factor. I could say that there are 1.3357899823324798x10^14 atoms in an Asterion, and if I used the conversion properly I could do all my work in Asterions and have them come out the same as if I used the standard mole. Of course, I wouldn’t really do it, as my professors would kill me. I could also spend my entire life doing everything in moles if it wasn’t for the fact that it’d be completely nuts.
As an aside, how much would a mole of moles (the animals) weigh?
It has no effect.
A mole of the eastern mole, Scalopus aquaticus, would weigh 4.5 x 10^22 kg. The earth weighs 5.97 x 10^24 kg.
You are absolutely right, Desmostylus. And it does have an effect. Just not a very big one. E = Mc[sup]2[/sup] applies to all energy. Hot water has a greater mass than cold water, too.
Chava
Except, of course, that you have 16 decimals in that number, meaning that somehow you’d have to get together a final 49/50ths of an atom to have one Asterion.