Atomic number
The atomic number of an element is the number of protons in one of its atoms.
The definition
atomic number = number of protons in one atom of the element
It is written as Z.
A sodium atom has 11 protons in its nucleus, so the atomic number of sodium is 11.
Why it identifies the element
Every atom of one element holds the same number of protons, and different elements hold different numbers. So no two different elements can ever share an atomic number.
That makes the atomic number a unique fingerprint. If you are told an element has an atomic number of 6, you have been told it is carbon — no other element has 6.
Electrons, and what happens to ions
A neutral atom holds as many electrons as protons, so the atomic number also tells you the number of electrons in a neutral atom of that element.
But during a chemical reaction an atom may lose electrons or gain them. Such a charged atom is called an ion, and an ion has more or fewer electrons than it has protons.
An ion keeps its atomic number
Reactions move electrons, never protons. The proton count does not change when an ion forms — so the atomic number does not change either, and the ion is still the same element.