Historical and scientifical outline
The idea that matter could be composed of minute particles is very ancient. Democritus, in V century B.C., already thought that matter was composed of atoms, indivisible particles.
However, ancient phylosofists' hypothesys, although they were based on reasonement, were not verified experimentally.
It was only after the introduction of the scientific method that the hypotesis that the matter could be composed of athoms was verified by seeing its consecuences.
Since 1700 physicians and chemists understood that substances believed simple were instead decomposable in simpler substances. For instance, in 1800 English chemist Nicholson verified that water, once believe an element, was decomposable in hydrogen and oxygen by electrolysis.
This discovery contributed to make scientists believe that matter was made of particles. Since that, particles of decomposable substances have been called molecules, particles of elements have been called atoms.
In the same years many researchs by physicians and chemists leaded to understand that the mass of the reagents and the mass of the products in a reaction are equal (law of conservation of matter by Lavoisier); that in composed substances proportions of elements are constant (Proust's law); that in reactions proportions between elemnts are expressed by integers generally not very big (Dalton's law), etc.
In 1802 Gay-Lussac verified that, if by decomposing a substance gases werw obtained, proportions between their volumes are constant if temperature and pressure are constant. This intuition was taken in 1811 by Italian chemist Avogadro, who enounced that equal volumes of different gases, with constant pressure and temperature, contain the same number of molecules. This law, called Avogadro's law, is one of the bases of modern science.
Since the application of the law of Proust chemists began to calculate mass of single atoms of elements in relationship between themselves. For example, from its application to water chemists deduced that oxygen has a mass 16 times bigger than that of hydrogen.
Because of the discovery that hydrogen was the less massive element chemists assumed its mass as unit of measurement for the atomic mass(amu = atomic mass unit). In recent times the discover of isotopes led to review this unit; now 1 amu is 1/12 the mass of Carbonium 12, a quantity of matter very similar but not equal to the mass of hydrogen.
From Avogadro's law we can take a constant called Avogadro's number that is the number of molecules in a mole of an element, that is in his relative athomic mass expressed in gramms. If we take a mole (2 g) of hydrogen we can observe that, at 0°C and 1 atm of pressure, it takes a volume of 22,4 litres. Because of the Avogadro's law, any gas takes the same volume takes also the same number of molecules. Because 22,4 litres are always taken by 1 mole of a gas (at 0°C and 1 atm), the number of molecules in 1 mole of any substance is always the same. Avogadro's number (usually called N) is this number; it's been calculated experimentally and is equal to 6,022 · 1023.