Some forms of radiation is dangerous to you and some forms are not dangerous to you. Sunlight, which is a form of electromagnetic radiation, is good for you in moderate doses (it causes your skin to make D-vitamin), but not in large doses (can lead to skin cancer). As a general rule, high energy radiation (waves with a short wavelength) is dangerous to you whereas low energy radiation is bad for you. Radio waves for instance, are so low energy that they cannot do anything to you. At the other end of the scale you have cosmic rays that come in from deep space and hits the earth. Cosmic rays have a lot of energy and when they shoot through your body they may rip up a few DNA molecules, potentially causing the cell to become a cancer cell. Luckily the earths magnetic field protects us from cosmic rays and we generally do not have to worry about them (astronauts do however).
In between radio waves and cosmic rays you find microwaves. These waves have the special property of causing H2O molecules to jiggle. Now jiggling a molecule is the same as heating it up, that is why microwaves are great for heating foods which always contains a fair amount of water. For the same reason you do not want to stick your head in the microwave oven (your brain contains a lot of water to). Mobile phones also communicate using microwaves and when you are using a mobile phone will heat your head a little bit. I want to emphasize little because it is a minuscule amount of heating compared to other sources, and there is no scientific reason to assume that the heating would cause any adverse effects whatsoever.
What about radioactivity? Radioactivity is the high energy radiation that is generated when larger atoms fall apart. In general radioactivity is bad for you because just like cosmic rays it can destroy DNA molecules, however, some forms of radioactivity (alpha particles) are only dangerous if you first eat them (this is because, unlike gamma radiation, they cannot penetrate the skin). For this reason you want to avoid radioactivity as much as possible. Having said that, there is no way that you can avoid radioactivity completely because atoms are falling apart all around us all the time. This informative chart shows how much radioactivity you get from various things you are more or less likely to encounter in your life. For example tonight, when I sleep next to my wife, a certain number of atoms in her body will break, radioactivity will be emitted, and I will receive 0.05 micro sieverts of radiation. You get a hundred times more (5 micro sieverts) if you visit the dentist and get a teeth x-ray, that is why the dentist leaves the room when the picture is taken. If this sounds bad though, consider that a transatlantic flight will give you a full 40 micro sievert, four times as much as a dental x-ray!
All of this pales in comparison to a regular CT scan, a standard technique in medicine that gives you a full 2 milli sieverts, that is 2000 micro sieverts. A chest scan (which I have taken) will give you 7 milli sieverts. Still even at 7 milli sieverts there is no proved risk of adverse effects, although it is not clear that there is no risk either. The lowest dose that has been clearly linked to cancer is 100 milli sieverts. To get radiation poisoning you need 400 milli sieverts and if you get 4 sieverts (4000 milli sieverts), it is usually fatal...