If the source of the ionizing radiation is a radioactive material or a nuclear process such as fission or fusion, there is particle radiation to consider.
Particle radiation is subatomic particles accelerated to relativistic speeds by nuclear reactions.
This would put a source of alpha particles inside your body, wreaking havoc by ionising atoms in nearby cells.
Because living cells and, more importantly, the DNA in those cells can be damaged by this ionization, exposure to ionizing radiation is considered to increase the risk of cancer.
Thus "ionizing radiation" is somewhat artificially separated from particle radiation and electromagnetic radiation, simply due to its great potential for biological damage.
Typical alpha particles (α) are stopped by a sheet of paper, while beta particles (β) are stopped by an aluminum plate.
Gamma radiation (γ) is damped when it penetrates lead.
A common source of ionizing radiation is radioactive materials that emit α, β, or γ radiation, consisting of helium nuclei, electrons or positrons, and photons, respectively.
Other sources include X-rays from medical radiography examinations and muons, mesons, positrons, neutrons and other particles that constitute the secondary cosmic rays that are produced after primary cosmic rays interact with Earth's atmosphere.
Radiation with sufficiently high energy can ionize atoms; that is to say it can knock electrons off atoms, creating ions.
Ionization occurs when an electron is stripped (or "knocked out") from an electron shell of the atom, which leaves the atom with a net positive charge.
The cell is then likely to do something very different to what it's supposed to do, for example, it may turn cancerous and start multiplying uncontrollably.
Thus alpha particles, whilst they have a low penetrating power, can be the most dangerous because they ionise so strongly.
While the part of the ultraviolet spectrum that penetrates the Earth's atmosphere is non-ionizing, this radiation does far more damage to many molecules in biological systems than can be accounted for by heating effects, sunburn being a well-known example.