Every second of your life your body is hit with about 15,000 particles of radiation. This radiation comes from all kinds of sources: potassium, bricks, outer space. Any one of these particles could kill a cell or damage it to the point where it goes berserk replicating itself.
That's how cancer begins. The chance that one of these particles will cause cancer is extremely low, so don't panic. Still, the more radiation you're exposed to over long periods of time, the higher the odds you'll develop the big C. The safety workers at the Japanese nuclear power plant are being exposed to very high levels of radiation.
So, how does it compare to what the average person is exposed to? For example, how many cigarettes would you have to smoke to have the same levels? How many bananas would you have to eat? How much TV would you have to watch? The answers are here.