Why are people so afraid of ionizing radiation? Mostly because we cannot see it, hear it, smell it, taste it, or touch it, and because we have heard scary things about it from movies or the news media or people we know. The key thing to remember is that the harm done by radiation depends strongly on the dose.
Nuclear radiation is not a binary hazard -- you measure it in milliSieverts. The more milliSieverts you accumulate in dose, the more risk. To learn how to count milliSieverts in a straightforward and engaging way, go here. If you would like to understand a little more about radiation, read on.
What are the key features of radioactivity? The first is that each atom in a radioactive sample of a given chemical element has a constant probability of undergoing nuclear fission per second. You cannot tell which particular atom is going to decay radioactively at any moment; it is up to purely random chance whether or not it does. Another important aspect of radioactivity is that radioactive versions of different chemical elements decay at different rates. Physicists characterize how fast a radioactive decay happens by its half-life, which is the amount of time it takes for a radioactive substance to decay to half of its former activity.
Now let us imagine comparing two different radioactive samples, say Iodine-131 and Plutonium-239. Which would you rather stand next to: a box of a zillion I-131 atoms, half of which will radioactively decay in the next eight days, or a box of a zillion Pu-239 atoms, half of which will decay in the next twenty-four thousand years? If you guessed the plutonium, you were correct. When samples of two radioactive sources contain the same number of atoms, the sample with the shorter half-life is the more dangerous one and the sample with the longer half-life is the less dangerous one. Something with an infinitely long half-life is harmless: it never decays radioactively at all.
If the samples do not have the same number of atoms as each other, the answer to the question of which is the most dangerous will shift. Suppose instead that we adjust two samples to start out with the same number of radioactive decays per second, i.e. they register the same number of clicks on a Geiger counter. In this case, the activity of the sample with the longer half-life will decay more slowly over time than the sample with the shorter half-life, because it has a longer half-life. This is why people often speak of Plutonium-239 as evil: it decays away reeeeeeally slowly, taking over twenty-four thousand years before the Geiger counter reading will drop by a factor of two.
It is important to recognize that we also need to know other things about our source of radioactivity than just its half-life to calculate its overall danger to humans over time -- and this is where the biology gets really important. The mechanism of exposure, the biology of the element, whether the radiation is of alpha beta or gamma type, and the radiochemical behaviour of the decay daughters all come into the equation. This is why being a radiologist takes a lot of training.