Step-by-step explanation:
Barry gave a great introduction below but failed to answer the actual question. Radioactive isotopes are those isotopes that are naturally unstable due to the number and configuration of their protons and neutrons. Inside a nucleus, there is a balance of forces going on and the two dominant ones are the strong nuclear force and the electromagnetic force.
First, the electromagnetic force. This is one that we are intuitively familiar with. If you try to push two magnets together (of the same polarity) they force each other apart. This same effect occurs between protons in the nucleus. They are extremely close together and are constantly trying to force each other apart. Since every single atom around is not violently flying apart, there must exist some force that counteracts that force…
Which brings us to the strong nuclear force. This force exists between all nucleons (protons and neutrons) equally and is a strongly attractive force. This attractive force acts to pull nucleons together but it has a strictly limited range. It can only affect its nearest nuclear neighbors.
This leads us to several conclusions: 1) if an atom has too many protons and/or not enough neutrons, the repulsive forces will eventually win out of the atom will split and 2) there is a point at which a nucleus becomes so large that the electromagnetic forces (that have an unlimited range but drops at the square of the distance) will always win out over the strong nuclear force that has a finite and very short range.
This is some of the foundational knowledge of nuclear physics. This is why there are a series of stable atoms at the lower end of the proton numbers (they can simply add neutrons to become more stable when the nuclei are smaller) and why after a certain point (lead, it turns out) all isotopes are radioactive and will eventually decay due to the nucleus simply being too large.
Now, for the second part of the question. Radioactive isotopes are used for many things and an exhaustive list would be, well, exhausting, so here are a few examples to get you started:
Nuclear medicine. Radioactive isotopes are ingested or injected and their radioactivity is captured as it exits to body to track things like flow rates, diffusion rates, or stay times within the body. They can also be used for imaging studies like in a Barium Swallow Test.
Non-destructive material testing. The radiation from isotopes can be used to test materials to determine certain properties without damaging the materials. Neutrons are also good for this but that involves putting your materials into an active research reactor.
Smoke detectors. They are make with small samples of alpha emitters. The alphas are produced at one end of the device and are detected at the other end. If smoke gets in-between the emitter and detector, the alphas are blocked and the system alarms.