The radiation from a nuclear power plant consists of subatomic particles traveling at or near the velocity of light. They can penetrate deep inside the human body where they can damage biological cells and thereby initiate cancer.
Radiation is indeed everywhere but only lethal doses of radiation of 500 rems can kill a person. However, 80% of radiation comes from natural sources, such a cosmic rays and Earth itself. The human body is a significant source of natural radiation, primarily from the potassium we ingest (Hewitt, 2008, p. 315). But the good news is that our cells repair most kinds of molecular damage caused by radiation if the radiation is not too severe.