Notably, the neutrons do not change the chemical behavior of the atom. So there are three isotopes of Carbon that can exist in nature. C-14, however, is unstable. Eight neutrons is just too much of a good thing when there are only 6 protons. The escaping alpha particle collides with something and again, the kinetic energy is deposited as heat.
Radon gas is radioactive, and radon itself comes from the decay of radioactive uranium.
The uranium is locked into rocks but when the uranium becomes radon, the radon, being a gas, bubbles out of the rocks and into your house. If you inhale the radon gas into your lungs, and that radon atom has the bad sense to undergo decay while in your lungs, that alpha particle will collide with your lungs. That could damage your DNA and, if you're unlucky, the resultant damage could produce cancer. Plutonium has the same problem. Before we get into the details of radioactive half-lives, let me give you some numbers to chew on.