Oxygen having the atomic number of 8 and an atomic mass of 16 must therefore have neutrons and electrons

The atomic number is the number of protons in the atoms of an element. The atomic mass is found by adding the number of protons and neutrons together. In this case, we know the atomic mass (16) and the number of protons (8) - the atomic number. 
Simply subtract: 16-8 = 8 Neutrons

The number of electrons in an atom equals the number of protons in order to have a charge of zero. Therefore, 8 electrons.
 (if the atom had a net positive or negative charge, it would have fewer or more electrons to result in the charge, and would be called an ion)