What causes atoms to be radioactive?

Atoms found in nature are either stable or unstable. An atom is stable if the forces among the particles that make up the nucleus are balanced. An atom is unstable (radioactive) if these forces are unbalanced… An unstable nucleus will continually vibrate and contort and, sooner or later, attempt to reach stability by some combination of means:

* ejecting neutrons, and protons

* converting one to the other with the ejection of a beta particle or positron

* the release of additional energy by photon (i.e., gamma ray) emission.

Read more here: http://www.epa.gov/rpdweb00/understand/radiation.html

If you would like to investigate this topic in a more advanced way, a good place to start is to search for information about the electric force between protons that tries to disintegrate the nucleus and the strong nuclear force that tries to hold the nucleus together.