(This is the topic of my dissertation, and I have also published a few articles related to it. I thought I'd try to explain it without using too much jargon for friends and family who may wonder what I've been doing with my life. :) Feedback is always appreciated!)
Stable isotopes of different elements have many interesting applications in areas such as geophysics and archaeology. While the identity of a given element is determined by the number of protons in the nucleus, the nucleus also contains a certain number of neutrons that help keep the nucleus stable. (When a nucleus has too many or too few neutrons, it will become radioactive.) These neutrons usually do not affect most of the chemistry of an element with notable exceptions. Variations in the number of neutrons do lead to different isotopes of each element with different masses, such as carbon-12 (12C), carbon-13 (13C), and carbon-14 (14C). The number next to the element is the total number of protons and neutrons, which is the atomic mass of that atom. The amount of each stable isotope on Earth stays relatively constant. For example, 13C makes up about 1.1% of all carbon on Earth. However, the differences in masses can cause small changes in the amount of each isotope in a chemical through different reactions or physical processes such as evaporation. Usually the differences between isotopes are related to their mass differences through atomic or molecular speeds in a gas or the spring-like vibrations of chemical bonds. These small changes in the amount of an isotope allow researchers to learn new information about the physical, chemical, or biological processes that affected an object.
While in almost all cases stable isotopes move and react in ways that depend on their mass, the three stable isotopes of oxygen (16O, 17O, 18O) in ozone formed in the laboratory or the atmosphere like to break the rules. Scientists expected that 16O would form ozone the fastest, 18O would form ozone the slowest, and 17O would form ozone at a speed about halfway in between. Instead, not only does 16O form ozone the slowest, but 17O and 18O form ozone at the same faster speed! To understand just how strange this is, imagine that you have three balls, one light, one heavy, and one in between. If you dropped them all from the same height at the same time, the heaviest ball would land first, then the ball in between would land, and followed finally by the lightest ball, right? What if the lightest ball landed first, and then the two heavier balls landed later, at the same time? That's a lot like what happens in ozone formation.
Scientists call this phenomenon "mass-independent", but it probably should be called "mass-bizarre". It's not necessarily unique to ozone, either. Scientists have discovered this strange isotope behavior in oxygen isotopes in some meteorites and in sulfur and mercury isotopes as well. More than a mere curiosity, the signal in the isotopes from these processes is unique and so could potentially be useful as a tool for understanding many different geophysical processes. This "mass-independent" signal finds its way into all sorts of other chemicals in the atmosphere, such as CO2. Despite its potential use as a tool to trace different processes in the atmosphere, the chemistry and physics of this weird effect remains poorly understood. Improving our understanding of what leads to these strange effects is key to using them to understand more about the Earth's atmosphere.