Isotopes exist because the atomic nuclei of many elements are stable or somewhat stable with different numbers of neutrons in them. Because the identity of an element is determined solely by its number of protons, and its chemical properties solely by its number of protons and electrons, different numbers of neutrons change neither the element nor its chemical properties. Nonetheless, many isotopes are radioactive and unstable, breaking down in time.
Radioactive isotopes are the most famous, both for their inherent dangers and their use in nuclear power and nuclear weapons. These isotopes are unstable, which means that over time, any particular sample of such an isotope gradually transforms into one or two other elements. This can occur because the atom splits or because a neutron becomes a proton or a proton becomes a neutron. All these processes involve the release of energy or energetic particles, and thus are radioactive. Most radioactive isotopes are so unstable that they are not found in nature, as they decay too quickly.
According to resources on the University of Oregon website, before 1906, it was generally thought that the mass of an element was integral to its identity. However, in 1906 and 1907, several researchers proved that there were elements of different atomic mass but identical chemical properties. It was Frederick Soddy, in 1913, who first coined the term "isotopes."