Every atom has a nucleus containing protons and neutrons. The number of protons in the nucleus is called the atomic number of the element. All atoms of the same element have the same number of protons, and atoms of different elements have different numbers of protons. This is why the atomic number identifies an element.
Atomic number also gives the position of the element in the periodic table. Hydrogen has atomic number 1 so it is the first element. Oxygen has atomic number 8, so it is the eighth element and each oxygen atom has 8 protons. Sodium has atomic number 11, so it is element number 11 and each sodium atom has 11 protons.
Do not mix up atomic number and mass number. The mass number is the number of protons plus the number of neutrons, so it is bigger than the atomic number for every atom except the most common type of hydrogen.