How are the atomic number and the number of protons related to each other? how do the number of protons, number of neutrons, and the mass number relate to each other? what is the one thing that determines the identity of an atom (that is, whether it is an oxygen atom or a carbon atom, etc.) ?

The atomic number (Z) uniquely identifies a chemical element. In an uncharged atom, the atomic number is also equal to the number of electrons.

The atomic number, Z, should not be confused with the mass number, A, which is the number of nucleons, the total number of protons and neutrons in the nucleus of an atom.

In this video Kristine Born explains this two concepts in more detail.