there are different types of radioactivity … but when they are damaging they are damaging in the same way – they are high energy particles that can penetrate your skin (normally light or other particles just bounces off our skin (that’s why not see-through!).
The safest type of radiation is alpha particles (these are basically the nuclei of helium atoms) – these are relatively slow and heavy and normally don’t penetrate our bodies and so are quite harmless.
Then you can get energetic electrons (beta radiations) or energetic light (from x-rays up to gamma rays) – these *can* penetrate your skin… the more energy, the further they go. As they go through your body they rip the electrons out of your atoms and this is bad for you – it makes different bits of your cells charged, and so things start sticking together that shouldn’t and repelling each other when they should stay together (like a cell wall) and so on… they can also knock bits out of your DNA or make your proteins fold badly.
In small doses your body can repair itself (whichi is why x-rays are generally ok, although you shouldn’t have xrays for no reason), but in big doses radiation can cause severe illness and even death.