The nuclei of the atoms of the various elements of the periodic table of elements are built up of protons and neutrons. The number of protons determine the charge of the nucleus and the element itself. The amount of neutrons can vary for the different elements. Atoms of the same element, i.e. the same number of protons, varying in the number of neutrons are called isotopes. The small and light elements have at least one, but in most cases several, stable isotopes. For example, the lightest element hydrogen has two known stable isotopes: protium, consisting of only one proton and deuterium, which has one neutron in addition. Tin is the element with the most stable isotopes. It has ten different ones.
In addition to the stable isotopes every element has radioactive ones that will decay by alpha or beta decay, or in some cases, spontaneous fission. Lead is the element with the heaviest nucleus that can still be considered stable. Any heavier element, e.g. uranium or plutonium, has no stable isotope meaning every atom is radioactive.
So the question arises, why are some atoms more stable (tin) than other (heavier) ones? Similar to the case with valence electrons, there are also shells in which the nuclei are arranged. When you think about the noble gases for example, why are they so stable and inert in a chemical way? It is because of their fully filled valence electron shells. When dealing with atomic nuclei you will find so called magic numbers: 2, 8, 20, 28, 50, 82, and 126. Elements having protons and/or neutrons in the amount of a magic number are more stable than their surrounding elements. The above mentioned tin for example has 50 protons. For neutrons the magic number 126 can be observed in (almost) stable isotopes of lead and bismuth. For protons the number 126 was not reached, yet.
Glen Seaborg was the first one thinking about an Island of Stability, which would arise at higher magic numbers as for example 184 and 258. It is up to nuclear chemists and physicists to produce those elements. Maybe those atoms are stable and long living enough to do fascinating chemistry with them.
 J. R. de Laeter, J. K. B?hlke, P. De Bi?vre, H. Hidaka, H. S. Peiser, K. J. R. Rosman, P. D. P. Taylor, Pure Appl. Chem. 2003, 75, 683-800.
 http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/shell.html (last access 06.10.2013).
 R. Herrmann, Physica A 2010, 389, 693-704.