Radiation damages DNA
Radiation acts by damaging DNA. When it hurts us, it damages DNA in healthy cells which are doing their job keeping us alive and well. When we use it in medicine, e.g. in radiotherapy, it acts by damaging DNA in cancer cells.
Radiation can damage DNA either by scoring a direct hit, or by breaking-up water. The broken water is very reactive and can cause damage to DNA (or anything else it comes across). Radiation comes in three forms, alpha, beta, and gamma.
Alpha particles are a ball made up of two protons and two neutrons - essentially a helium atom, a large positively charged blob.
Beta particle is a nerdy name for an electron - a small negatively charged blob - it is nearly 1/8000 times the size of an alpha particle.
A gamma ray is just like normal light, but very energetic. Like UV light you can't see it, but like UV light it burns.
Radiation damages DNA - cells work hard to repair it.