What exactly is a mole? I have some trouble understanding exactly what a mole represents. As I understand, one unit mole is 1/12 of the mass of an atom of carbon-12 (thus it is the mass of one nucleon?). What is a mole, then?
 A: The mole is of such great use because of the predictable and unique relationship between the number of atoms (counted in moles) in a sample and the sample's easily measured weight or mass (depending on your unit system). 

The mass per mole of a substance is called its molar mass. Since the
  standard unit for expressing the mass of molecules or atoms (the
  dalton or atomic mass unit) is defined as 1/12 of the mass of a 12C
  atom, it follows that the molar mass of a substance, measured in grams
  per mole, is exactly equal to its mean molecular or atomic mass,
  measured in daltons; which is to say, to the substance's mean
  molecular or atomic weight.
  -Wikipedia

For example: one mole (6.02X10^23 atoms) of Carbon atoms has a mass of 12.011 grams, which is also the atomic weight of Carbon in Daltons (shown in this periodic table). The molar mass of water (2 hydrogen atoms and 1 oxygen atom) =  2*(1.0079) + 1*(15.999) = 18.0148 grams (I am using a lot decimal places for effect despite that being impractical in real life). 
A: I mole is just a way to represent a number of atoms. Instead of saying 12 eggs we can also say a dozen. Similarly one mole of carbon is the same as saying $6.02\cdot10^{23}$ atoms of carbon.
A: A mole is nothing more than a countable number of things.  Specifically, it is approximated by 6.02 × 1023 number of things.  Hence, you can have a mole of argon atoms, a mole of electrons, or a mole of house keys all of which contain approximately 6.02 × 1023 of their respective items.  Note that each of these groups of things will have a different mass.
What you are referring to as 1/12 of a 12C atoms is not a mole, but an atomic mass unit.
A: A mole is like saying "a dozen," as Luksen points out in a comment.  But why pick $(6.02)(10^{23})$ instead of, say, $(6.07)(10^{24})$?  The reason is that the number was strategically picked such that the weight, in grams, of a mole would be numerically exactly the same as its molecular weight.  It is a number that was formulated for the convenience of the chemist (just like the "dozen" was created for the convenience of... uh... someone... the baker?  No, that was 13...)  
So, for instance, if you know you have a mole of a substance, you can weigh that mole, in grams, to find the molecular weight without doing any math (like 18.0 grams is about a mole of water, which has a molecular weight of about 18.0).  If, on the other hand, you want a mole of a substance and you know its molecular weight (as you almost always do) you can weigh it out in grams to get exactly a mole.
