How do protons and electrons determine the properties of an atom?

They determine if it has a negative or positive charge

Well protons will contribute to what we all know as the mass of an atom, the number of protons always determine the exact identity of whatever Element you want to describe or find more about. The Electrons always have a negative charge, But because the atom is electrically Neutral, that must mean that the exact number of the Electrons will always equal up to the number of the Protons.