The island of stability is a term from nuclear physics, which describes the possibility of elements which have particularly stable "magic numbers" of neutrons. This would allow certain transuranic elements to be far more stable than others, and thus decay much more slowly.
The idea of the island of stability was first proposed by Glenn T. Seaborg. The hypothesis is that the atomic nucleus is built up in "shells" in a manner similar to the electron shells in atoms. In both cases shells are just groups of quantum energy levels that are relatively close to each other. Energy levels from quantum states in two different shells will be separated by a relatively larger energy gap. So when the numbers of neutrons and protons completely fill the energy levels of a given shell in the nucleus, then the binding energy per nucleon will reach a local minimum and thus that particular configuration will have a longer lifetime than nearby isotopes that do not have filled shells.
The magic "number" of neutrons is 184, and the matching proton numbers are 114, 120 and 126 — which would mean that the most stable possible isotopes would be ununquadium-298, unbinilium-304 and unbihexium-310.
The term "particularly stable" is in comparison to the half-lives of slightly lighter or heavier elements; the half-lives of elements in the island of stability are still expected to be measured in fractions of a second, or perhaps measured in days, though some theoretical possibilities include much longer periods.
See also