In grad school, we have to take a lab safety class. One of the topics is radiation. It's actually really simple.
Atomically unstable materials give off radiation as they decay. If you're close to some, the emitted radiation goes into you and starts to knock around your molecules. As soon as you move far enough away or turn off the source, as with an X-ray machine, you stop receiving radiation. Radioactivity is not contagious so just from being around radioactive material will never make you 'glow'. Even a huge, deadly dose of radiation won't turn you radioactive in the hours before you succumb.
What is bad is contamination. If one were to touch radioactive material or breathe in dust from radioactive materials exposed to air, then some of the dust gets embedded in one's body. Assuming you can't remove the individual atoms, those atoms continue to emit radiation from inside your body and whatever body part is then effectively radioactive. That's why people dealing with nuclear materials wear spacesuits and get scrub-downs.
With this in mind, nuclear bomb blasts aren't necessarily terrible--if you overlook the horrific burns and destruction due to high pressures, winds, and temperatures--but the fallout is. Similarly exhaust from burning coal can be an issue since burning it releases very small amounts of radioactive atoms into the air, in addition to chemicals like mercury. Lastly, you might have heard of dirty bombs. Those are normal bombs packed with radioactive dust so when it is exploded, the dust is scattered.