A decay chain occurs when an unstable isotope can not form a stable isotope in a single decay, and thus must go through a number of decays before reaching stability.
Decay chains are mostly present in very heavy isotopes. For the most part, most lighter isotopes, whether they are produced by technology, cosmic ray spellation, or (in the case of Potassium-40) because they are primordial, are fairly close to stability. They undergo a single decay, almost always a beta decay, and turn into a stable isotope.
The main elements that have long (and often complicated and branching) decay chains are the long-lived, but still unstable actinides. Specifically, uranium, thorium and plutonium. For reasons that I don't understand and aren't currently important, the isotopes of these elements are on an island of stability, and most of the elements slightly lighter than them are all unstable, some with very short half-lives. Their decay chains carry them, from their precarious perch, quickly through these elements, to the complete stability of lead.
Decay chains are important for two reasons, having to do with their absence and presence. In the lighter elements, their absence makes the technological use of isotopes, in medicine particularly, much safer than it would otherwise be. For example, Fluorine-18, which has a half-life of two hours and is used in medical testing, and emits a positron, turns into Oxygen-18, which is completely stable. If this isotope had a complicated decay chain, with varying products of varying half lives, its possible physiological consequences would be much more dangerous and unpredictable.
The presence of decay chains in heavier elements is important for two reasons. First, it means the total energy released is much higher than it would first appear. The alpha decay of an atom of Uranium-238 releases 4.5 MeV of energy. However, accounting for all the decays in its chain before it becomes a stable isotope, it actually releases ten times that amount of energy. It also, during that decay chain, passes through some elements with quite different chemical properties, such as when it becomes Radon gas, which can be inhaled. In summary, the additional energy and variance in chemical properties of the elements in a long decay chain of the actinides make radioactivity much more dangerous than it would otherwise be.