An atom's identity is determined by the number of protons it has in its nucleus. This number is referred to as the atomic number, which defines the element. For example, hydrogen has one proton, and its atomic number is 1, while carbon has six protons and an atomic number of 6.
The number of electrons and neutrons can vary within atoms of the same element (leading to ions and isotopes, respectively), but it is the number of protons that defines the identity of the atom.