A Nitrogen atom with 7 protons and 8 neutrons has a mass number of 15 amu. However on the periodic table, the atomic mass for Nitrogen is 14.01. How is this possible?

Because there exist nuclei with with 7 protons and MORE OR LESS than 7 neutrons; that is nitrogen possesses a number of .

If a nucleus possesses 7 protons, 7 fundamental positive particles, it is by definition a nitrogen atom. The most abundant isotope (as they are called) is ##""^14N##, however, there exist isotopes where the nitrogen nucleus contains more than 7 neutrons. If there are 8 neutrons, fundamental neutral particles, we have the isotope ##""^15N##. This isotope is still nitrogen (why? because it has 7 protons) but it masses more than the regular isotope.) and its chemistry is still the same. The of nitrogen is thus the weighted average of the individual isotope.

##""^15N## is a very important isotope, because it is spin active, and you can perform NMR experments with it, just as you can with the ##""^1H## nucleus. Inorganic chemists will buy isotopically labelled nitrogen for precisely this reason (it costs an arm and a leg tho!).