Hydrogen being the lightest element was assumed to have mass for 1 atom equal to one atomic mass unit (a.m.u). The number does not signify the mass of an atom in grams. It is just a pure number.
The mass of atoms of other elements was compared to that of hydrogen, in order to find the relative atomic mass. If one atom of sodium weighs as much as 23 atoms of hydrogen, then the atomic mass of sodium is 23 a.m.u.Atomic mass unit
Old definition
The mass of one atom of hydrogen taken as 1 is called the atomic mass unit.
Drawback of using hydrogen as atomic mass unit
Hydrogen gas in its natural state has 3 isotopes of atomic mass 1, 2 and 3 respectively. Thus average mass works out to be 1.008 a.m.u rather than 1 a.m.u. This in turn complicates the atomic masses of all other elements.
Changes
Later on, an atom of oxygen was preferred as standard by taking its mass as 16 units. In 1961, the international union of chemists selected the most stable isotope of carbon (C-12 isotope) as standard for comparison.
For example, if 1 atom of Na weights as much as 23 parts of 1/12 of 12C isotopes, then the atomic mass of sodium is 23 a.m.u.