Stable isotopes are those which make up most matter; the isotopes which do not decay radioactively, or have an extremely small decay constant. Most chemical and physical situations do not distinguish considerably between different stable isotopes of the same element, but there are several branches of science built up around the study of isotope ratios. The process whereby these ratios change is called "isotope fractionation", and this finds the most applications in stable isotope geochemistry.
Generally speaking, stable isotopes are clustered around some norm, and isotopes with far fewer or more neutrons become less stable. Isotopes with a magic atomic mass or atomic number are particularly likely to be stable (yes, that's the technical term, I'm not being facetious).