Nuclear radiation is basically subatomic particles such as alpha (2 protons 2 neutrons or a He ion w/o electrons), beta particles (electrons), and gamma radiation (electromagnetic wave). Because alpha particle are largest, they generally does not penetrate very far, thick clothing such as jeans generally keep most of these particle from entering your body. Beta particle are smaller thus require a bit more protection (sheet of wood). Gamma radiation are the most difficult to contain, they generally require thick sheet of lead to keep at bay.
Each of these particles carry very low energy, but at the microscopic level, they can cause massive damage. If one of these particle hit a molecule in your body such as your DNA, it can knock something off and cause a mutation. At the most extreme, this can lead to cancer or massive cell destruction (death).
While yes, nuclear radiation is bad for you, IT IS DEPEND ON DOSES. The mechanism described above only cause problems when the damage overwhelm your body natural healing ability. At anytime and anywhere, there is a constant background radiation that everyone is exposed to. It is at higher doses than background for a long enough duration that problems begin to developed. Ex: If I spend 10 min touring a nuclear test site, I would not likely to develop problems compare to someone who work there 5 days a week.