The Mass Number (A), is the total number of protons and neutrons in an atomic nucleus.
The Mass Number gives the approximate weight of the atom, because it counts the weight of the protons and neutrons, does not include the electrons weight because they are negligible. (See Atom Description for the values)
The Mass Number, does not identify an atom or element, in fact any element has different values of Mass Number. For example, the Hydrogen (Atomic Number, Z=1) has three values, A= 1, 2, 3.
Mathematically, the Mass Number is expressed as, A = Z + N
Where, N= number of neutrons
Thus, the Hydrogen has three Mass Numbers because some of them have zero neutrons, others have one neutron, or they could also have two neutrons.