|Map of stable isotopes|
The number of neutrons in an atom's nucleus determine what isotope you have of any particular element. The N axis along the left side of the graph indicates the number of neutrons.
Each colored dot plotted in the above graph represents a particular isotope of a particular element.
Isotopes have different degrees of stability as measured by their half-life. This is indicated by the color of the dot. The key for the colors is listed in the vertical bar to the right of the graph. Black indicates that isotope is stable, white indicates it is unstable. Near the top we have 10 to the 14th power years, which is one hundred trillion years. Near the bottom we have 10 to the -8 power seconds, which is ten nanoseconds.
I suspect that half-lives are measured by the amount of radiation a given mass of an isotope produces. Even a relatively small chunk of matter is going to have trillions of atoms. If your chunk of matter is composed of an isotope with a very long half-life, then in any specific length of time very few atoms will break up, but those break ups will produce radiation which can be measured and so from that data we can estimate the half-life.
Via Detroit Steve