Radioactivity is the conversion of one nuclear state to another. It is accompanied by the emission of particles or electromagnetic radiation. The nuclear state change can often involve a change in the atomic number of the atoms involved. The most well known forms of radiation emiited are alpha particles (helium nuclei - two protons bound with two neutrons), beta particles (energetic electrons) and gamma radiation (very high energy electromagnetic radiation). Another form of radioactivity is fission in which a nucleus splits into two smaller components.

Most sources of natural, predictable radiation come from the decay of atomic nuclei, resulting in either alpha - $\alpha$ or beta - $\beta$ particles. In general, $\alpha$ decay is more common among the heavier elements, as it reduces the proton:neutron ratio, while $\beta$ decay is much more prominent among lighter elements, as it converts a neutron into a proton.

Two prominent examples of radioactive decay are:

$\boldsymbol{\alpha}$ decay:

$U^{238} \longrightarrow Th^{234} + He^{4}(\alpha)$

This is the first decay in the famous Uranium decay. U-238 is essentially non-radioactive (especially compared to hyper-active U-235), and has a half-life of over four billion years.

$\boldsymbol{\beta}$ decay:

$C^{14} \longrightarrow N^{14} + e(\beta)$

This is the decay that allows for carbon dating, and has a half-life of over 5000 years.

Gamma radiation is much more difficult to come by, as emitting a gamma ray does not allow an atomic nuclei to decay. The most famous source of high-energy gamma rays is what happens when an electron and a positron annihilate:

$e^{+} + e^{-} \longrightarrow \gamma + \gamma$

Since positrons are relatively rare, this is an interaction that is relatively hard to find. However there are fairly reliable sources of $\gamma$ radiation, including Cesium-137 and Cobalt-60. Both are useful for a wide variety of technical purposes, as well as for their utility in cancer treatment.