The average person in the United States receives about 360 mrem every year whole body equivalent dose. This is mostly from natural sources of radiation, such as radon. In 1992, the average dose received by nuclear power workers in the United States was 3 mSv whole body equivalent in addition to their background dose.
Radiation causes ionizations in the molecules of living cells.
These ionizations result in the removal of electrons from the atoms, forming ions or charged atoms. The ions formed then can go on to react with other atoms in the cell, causing damage. An example of this would be if a gamma ray passes through a cell, the water molecules near the DNA might be ionized and the ions might react with the DNA causing it to break. At low doses, such as what we receive every day from background radiation, the cells repair the damage rapidly. Most cells that die are of little consequence, the body can just replace them.