Radiation: what is the risk ?

Natural radioactivity is found everywhere in nature. We can use it to measure time, because each radioactive isotope decays at its own rhythm. Archeological artifacts, as well as the age of the earth, the age of a cave, the age of ice, the age of human ancient bones can be deduced by measuring their radioactive activity. The yearly levels of natural radioactivity for humans (2.4 mSv) pose no risk: there is life on earth. In addition we take on average 0.5 mSv a year from medical imaging radiation. However, if you build a house in granite (usually rich in uranium) and if it is well insulated, you might have a problem: you will need to ventilate enough to get rid of the radon gas constantly produced in your house.
When radioactivity is used by scientists and engineers, it is a different ballgame, because there is more radioactivity concentrated in a smaller space. Pierre and Marie Curie, for instance, suffered burns from some of their radioactive samples.

Left: a page of the Curie's notes; Right: autoradiography of the same page 

showing a thumbprint (lower right) and some radium deposits on top.

(Musee Curie, Paris)

The dangers have been identified for over a hundred years. I found over the years that students who work with radioactive objects (we call that a "source") react in different ways. They range from the person who is too afraid to take on the job to the person who is too "macho" to respect the rules of caution. You cannot be macho with radioactivity: you cannot see it or smell it, but just like bacteria, it can kill. 
So, a biology prof. will have designed for the students experiments that include very little risk, but watch carefully for the clown of the class who never washes his hands; a radioactivity prof. will have experiments offering very little risk, but watch for the clown who does not follow procedure and eventually exclude him from the class (in my experience, it is always a male).
Workers who work in nuclear facilities can take a max dose of 50 mSv per year. However, the rule is not that nobody cares until they reach that dose, the rule is that one should achieve occupational doses that are as low as is reasonably achievable (ALARA). As a result, about half of the workers on nuclear plants present no measurable radioactivity over background, and the others typically have doses of 1 to 2 mSv.   At doses one thousand times stronger, you got very very serious health problems.
What are the rules? Pretty simple:
1. Limit the time exposure. Do not stay close to the source any more than you have to (if you work with one source, why keep several close to you? Put them away!)
2. Use a shield to protect yourself. Depending of the radiation level, you can handle the source with prongs without touching it; wear protection, like a lead apron; or keep the source behind a lead wall or in a special container. Each level of radiation requires an appropriate procedure.
3. Keep your distance. At two times the distance, the radiation is four times lower (because of the dispersion from the source, it is called the inverse square law).
So, if distance is so effective, how is it that one can measure traces of radioactivity all over the world after a nuclear test (like the US and the USSR did in the 50s and 60s), or an accident like Chernobyl? That is because small particles of radioactive material are scattered throughout the earth's atmosphere by winds. These particles go all around the world in less than a week. Soon after the Chernobyl accident, radioactive particles fell on the ground with rain as far from Ukraine as Finland and Greece. The spots of radioactivity on the following map are linked to rain events.

Image from http://maps.grida.no/

The radioactivity level was in such spots about 100 times the natural level. The fallout is made of particles, so the best way to be decontaminated is to take a shower. Is that all? No, because particles falling on soil will be integrated in plants and contaminate the food chain.  Then, it can be dangerous, because a radioactive particle which is absorbed by your body will keep giving radiation for all the time it is radioactive.

Why is it that scientists think that there is no danger in the US from a Japanese radioactive fallout? First because they evaluate the potential danger as lower than Chernobyl, but mostly because the US is far from Japan. On the following Peters map, you can compare directly the surface of Europe to the surface of the Pacific ocean: you could enter western Europe about two times between Japan and the US.