Chemists use the mole unit because, in chemistry, it is often important to speak in terms of number of atoms or molecules rather than in terms of mass or volume. However, it is difficult and inconvenient to directly speak in absolute number of atoms. The mole also provides an easy base from which to convert to units of mass and back again.
To know why chemists use the mole as a preferred unit of measurement in many contexts, it helps to have a firm understanding of what a mole is. One mole is the amount of substance that contains the same number of elementary particles, such as atoms, molecules or ions, present in a 12-gram sample of pure carbon-12. This number, 6.022 x 10^23, is also known as Avogadro's number.
A mole gives chemists a convenient way to speak of the stoichiometry of chemical reactions. Chemical reactions have stoichiometries based on the proportions of atoms and molecules involved, and because different elements and molecules have different atomic weights, thinking of reactions in terms of mass necessarily fails to give the full picture. However, because atoms are so small and there are so many of them involved in any reaction, talking about atoms in absolute terms is unwieldy. It is much easier to say that 2 moles of hydrogen react with 1 mole of hydrogen gas to form 2 moles of liquid water than it is to say the same thing in terms of mass or precise number of atoms, and it conveys more information, too.