The identity of the atom is, without going into too much detail, determined by its associated neutrons, electrons and protons. I have a feeling that this is the question you're asking though, it seems a little vague.
If you mean "What property determines which element an atom is?" then it's all about the atomic number, Z, which is the amount of protons in the atom. Atoms can vary within the element by means of loss/gain of electrons (Ionization) or by loss/gain or neutrons (Isotopes) however, if you add or remove a proton, the atom becomes a different element.