The following is a vastly over-simplified discussion of why stable isotope research works. If you are a stable isotope researcher, the imprecision of the description will drive you mad. But it is a fairly accurate description of the natural processes which are being used by researchers in so very many interesting ways these days. A more precise description of this process is provided in the article by Nikolaas van der Merwe called the Isotope Story.
Forms of Stable Isotopes
All of the earth and its atmosphere is made up of atoms of different elements, such as oxygen, carbon, and nitrogen. Each of these elements has several forms, based on their atomic weight (the number of neutrons in each atom). For example, 99 percent of all carbon exists in the form called Carbon-12; but the remaining one percent carbon is made up of slightly different forms of carbon. Carbon-12 has an atomic weight of 12, which is made up of 6 protons and 6 neutrons. The 6 electrons don't really count towards the weight, because they're so light. Carbon-13 still has 6 protons and 6 electrons, but it has 7 neutrons; and Carbon-14 has 6 protons and 8 neutrons, which is basically too heavy to hold together in a stable way, so it is radioactive.
All three forms react the exact same way—if you combine Carbon with Oxygen you get Carbon Dioxide, no matter what the number of neutrons. In addition, Carbon-12 and Carbon-13 forms are stable—that is to say, they don’t change over time. Carbon-14, on the other hand, is not stable, but instead decays at a known rate—because of that, we can use its remaining ratio to Carbon-13 to calculate radiocarbon dates, but that’s another issue entirely.
The ratio of Carbon-12 to Carbon-13 is constant in earth’s atmosphere. There are always 100 12C atoms to one 13C atom. During the process of photosynthesis, plants absorb the carbon atoms in earth’s atmosphere, water, and soil, and store them in the cells of their leaves, fruits, nuts, and roots. But as a result of the photosynthesis process, the ratio of the forms of carbon gets changed as it is being stored. The alteration of the chemical ratio is different for plants in different parts of the world. For example, plants that live in regions with lots of sun and little water have relatively fewer 12C atoms in their cells (compared to 13C) than do plants that live in forests or wetlands. This ratio is hardwired into the plant’s cells, and—here’s the best part—as the cells get passed up the food chain (i.e., the roots, leaves, and fruit are eaten by animals and humans), the ratio of 12C to 13C) remains virtually unchanged as it is in turned stored in bones, teeth and hair of the animals and humans.
In other words, if you can determine the ratio of 12C to 13C in an animal's bones, you can figure out what kind of climate the plants it ate during its lifetime came from. The measuring takes mass spectrometer analysis; but that’s another story, too.
Carbon is not by a long shot the only element used by stable isotope researchers. Currently researchers are looking at measuring the ratios of stable isotopes of oxygen, nitrogen, strontium, hydrogen, sulfur, lead, and many other elements that are processed by plants and animals. That research has led to a simply incredible diversity of human and animal dietary information.