what on earth is a mole, and why is it used in balancing equations

Well... chemistry is the study of very tiny particles (electrons, atoms, molecules). It would be very hard for us to work with, and make calculations with, a single particle. A 'mole' is just 6.022x10^23 of anything . E.g, 1 mole of carbon is just 6.022x10^23 carbon atoms. 1 mole of rocks is just 6.022x10^23 rocks. this idea of assigning a word to a number is used in day to day life. For example; a 'dozen' just means 12. (e.g a dozen rocks means 12 rocks)
The mole is incredibly useful for chemists. For example, it can tell us the ratio in which elements are reacting! For example, if you reacted pure nitrogen and pure hydrogen, 1 mole (or 6.022x10^23 atoms) of nitrogen will react with 3 moles (or 3 x 6.022x10^23 atoms) of hydrogen to produce ammonia (NH3). i.e every 1 atom of nitrogen will react with 3 atoms of Hydrogen! We use these when balancing equations.e.g N2 + 3H2 --> 2NH3i.e 2 nitrogen atoms have reacted with 6 hydrogen atoms to make 2 ammonia atoms. Therefore, the ratio of reaction of 1 nitrogen atom to 3 hydrogen atoms.