[Off-topic] xaccers:
Ionization is the process of adding or removing electrons from an atom, making it an ion.

So if you take a hydrogen atom which has no charge (1 proton, 1 electron) and knock the electron away, it becomes a positively charged ion.

Electrons orbit an atom's nucleus at specific distances, known as shells, and these correspond to energy levels.

When a photon of the right wavelength hits an atom it energises the electrons, in most cases it moves them into a higher energy shell, and when it returns it releases a photon equivalent in energy to the difference between the shells it has moved from. This is how we use emission spectroscopy to work out what elements are in a substance.

If the photon is of sufficient energy/wavelength it is able to energise the electron enough to leave the atom, which results in the atom being charged which is an ion.

Ions readily react chemically which is why it damages DNA as the molecules lose elements.

Photons have to be of a small enough wavelength to match the energy levels of the electrons, and the largest wavelength that can do this coincides with the part of the electromagnetic spectrum we call Ultra Violet.

The Ozone layer was mentioned earlier, this is made up of Oxygen in it's stable O2 form and less stable O3 form (which is what Ozone is).

UV radiation is able to break apart O2 molecules allowing a single oxygen ion to pair up with another O2 molecule making O3.

UV also splits up O3 into O2 and O, so the cycle continues. The Oxygen and Ozone absorb the UV photons that hit them which is why the split up. This greatly reduces the amount of UV radiation which reaches us on the surface but doesn't stop it all.

The remaining radiation is able to penetrate our skin and can lead to damage to our DNA which can result in cancer.

Cancer is where a cell divides and doesn't know when to stop.

Other, non-electromagnetic radiation which has been mentioned are alpha and beta particles.

An alpha particle is released through nuclear decay, it is basically a helium nucleus (2 protons, 2 neutrons).

When a nucleus of an atom is unstable it breaks down and ejects part of itself to become a more stable different element. It is relatively speaking huge, and only able to travel a few cm in air before hitting so many air molecules knocks it's momentum out. Tissue paper can stop it, as can a few layers of skin.

Beta particles are a free flying electron (or positron). They are able to penetrate much further than alpha particles, although a few mm of aluminium will stop them.

Imagine your DNA the the painting of the Mona Lisa.

Shoot it with a shotgun from a fair distance away, and it does a lot of damage, but each bit of damage is tiny, so you can still see it's the painting of the Mona Lisa.

That's electromagnetic radiation damaging your DNA. A low dose will damage various bits of it, but it can be repaired, you'd need a higher dose (equivalent of shooting a shotgun several times at Lisa) to do serious damage.

Now instead of a shotgun, use a 30mm round from an Apache helicopter, it blows a big hole in the painting, and depending on where it hits it may make it difficult to see the painting is of Mona Lisa. That's a beta particle.

Use a cannonball and pretty much no matter where it hits the painting it will make it difficult to tell what it is. That's an alpha particle hitting your DNA.

This is why you can survive a certain number of x-rays a year, or a certain number of high altitude flights, or a certain length of time out sunbathing, because the damage the radiation you're exposed to does is so widespread that it is easy for your body to fix.

Breathe in or swallow an alpha or beta emitter and it can potentially sit there smashing away at the DNA in the cells around it causing irreparable damage.

This however can be taken advantage of and used medically, for instance radioactive iodine can be used to scan (using a gamma camera) someone's thyroid gland, or damage it if it is over active. Likewise radiotherapy is used to treat cancers by damaging the cells further, as the healthy cells around the tumour should be able to recover, but the already damaged cancer cells should die off.