Note that radio waves are electromagnetic in nature as are tv waves (anyone remember UHF and VHF?). They are relatively safe because the wavelength is extremely long and the frequency is low. Penetration depth is inversely proportional to the energy of that wave. Radio waves don't tend to penetrate obstructions very well. Radar is not far away. You might have thought that radar is this high-tech penetrating radiation, but it is not so. Radar is ideal in applications such as military apps since it tends to bounce off objects rather than penetrate them. The bounce-back is what tells an operator that something is there.
Moving down the energy spectrum we come to infra-red radiation, which is the way some animals see (snakes being one example). After that, it's visible light, which is slightly penetrating (ever put a flashlight under your hand to see the bones slightly?). 
Then the UV bands. UV gets dangerous. It penetrates the skin to such a degree that the skin reacts. There are actually 3 bands of UV - UVA, UVB and UVC. The atmosphere and ozone layer filter out the UVB and UVC. You get a suntan (or burn) from UVA. UVB and UVC have more energy as they are higher in frequency and will begin to destroy the DNA molecule. That's bad. Fortunate that we live on a planet with such a well-designed atmosphere, no?After the UV spectrum, we get into microwaves and x-rays which have very high penetrating depths. X-rays are familiar to most that have ever been to a hospital. X-rays in particular have so much energy (because of the very high frequency and very short wavelength) that they go right through you, and impact a film behind you that then gives a picture of your insides. Note that in all hospitals, the x-ray room is shielded and only you are present for the pic, probably donning a lead vest. Smile!
More energetic than x-rays are gamma rays that are typically produced by nuclear decay and nuclear reactions, including fission. It is the most energetic type of radiation (save the mysterious cosmic rays) and thus has the potential to do the most harm. But that doesn't mean that it will hurt you. That depends on the "dose" you get.
A "dose" is the total amount of radiation you are hit with over a period of time. From the American Nuclear Society:
We live in a radioactive world - humans always have. Radiation is part of our natural environment. We are exposed to radiation from materials in the earth itself, from naturally occurring radon in the air, from outer space, and from inside our own bodies (as a result of the food and water we consume). This radiation is measured in units called millirems (mrems).
The average dose per person from all sources is about 360 mrems per year. It is not, however, uncommon for any of us to receive far more than that in a given year (largely due to medical procedures we may undergo). International Standards allow exposure to as much as 5,000 mrems a year for those who work with and around radioactive material.
Click the above link for an interactive chart that will help you gage the dose that you get. So 360 mrems is the average. A worker at Fermi 2? 310 mrems. Not bad! In fact, I just took a tour last week of the reactor building and stood no more than 30 feet from highly toxic spent nuclear fuel and the nuclear reactor core. My total dose for that tour? 0.4 mrems. How was this measured? Well, when you go into radiologically-active buildings, you go through a process, part of which is the issuing of an ED (electronic dosimeter) that continuously monitors your exposure. You also carry around a TLD (thermoluminescent dosimeter) with you at all times. All of your radiation exposure is continuously logged. Pretty cool!
UPDATE (3/20/11):Via Instapundit: RADIATION DOSES EXPLAINED at XKCD (click for bigger pic)
Previously:
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