The atoms of all elements (other than hydrogen) are made of three constituent particles, electrons, protons and neutrons (A hydrogen atom does not have any neutrons). The atomic number of an atom is equal to the number of protons it has while the atomic weight is the sum of protons and neutrons that are present in the atom.
Different isotopes of an element have the same number of protons but differ in the number of neutrons. Atoms of different isotopes of an element have the same atomic number though their atomic weight varies.
An atom of carbon has 6 protons. This gives an atomic number 6. The number of neutrons in atoms of different isotopes of carbon varies between 5, 6, 7 and 8. The most common isotope of carbon is C-12 which has 6 protons and 6 neutrons. The other isotopes have 5, 7 or 8 neutrons though the number of protons in all of them is the same. All isotopes of carbon are alike in that each atom has 6 protons.
Isotopes of an element differ in the number of neutrons contained in the nucleus of the atom. As such, the number of protons of the atoms stays the same regardless of the carbon isotopes; otherwise, the atom would no longer be considered a carbon atom. Because the only thing that is changed is the number of neutrons, the number of electrons and their placement stays the same as well. However, the atomic mass (displayed at the bottom of each rectangle in the periodic table) will differ among the isotopes because the neutron count matters for atomic mass calculations. That being said, if there are different common isotopes of a particular atom, the atomic mass listed is most likely to be a weighted average so it won't be perfect for any one isotope.
The definition of an isotope is atoms of the same element with the same amount of protons, but a different number of neutrons. This means that each carbon isotope has the same amount of protons. The element carbon, according to the periodic table, has 6 protons, so that means every other isotope also has 6 protons. The difference between the isotopes would be the number of neutrons which can vary.