Radiation takes three main forms. Alpha, beta, and light (uv up to gamma). These are helium nucleus, electrons, and photons respectively. Effectively a radio active isotope or fusion reaction will emit these and then as they fly through space they can kick electrons off molecules or even make whole atoms unstable. If this happens to DNA it will cause damage. Your body can’t repair all of it so it will accumulate and cause cancer. Your body mostly fixes it by killing all the damaged cells. This is why you get sun burns. Sun burns are mild radiation damage. If you get too much as a time you get organ failure as the cells of your organs die off to the point of them no longer being able to work. This is radiation poisoning.
Radiation of higher energy levels (so ultraviolet or higher) are able to ionize atoms. That means it can break apart molecules, including our DNA or just essential parts of cells, killing them or turning them to cancer.
Lower energy levels (per photon) but large amounts of radioation can harm us too by heating up water. For example a microwave uses radiation to cook food.
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