Two very different things, electromagnetic waves and high energy subatomic particles, are both described by the term 'radiation'. This leads to a substantial amount of popular confusion, as exemplified in the term 'nuke' as applied to the use of a microwave oven. Microwave ovens cook with relatively low frequency electromagnetic radiation. This is very different from any of the types of radiation generated in nuclear processes. More harmfully, many people are concerned with the possibility of contracting cancer through exposure to the low frequency electromagnetic waves emitted by electrical power lines and cellular telephones. There is no plausible scientific mechanism for any such effect, and meta-analyses of the studies supporting such fears have disconfirmed threatening findings as definitively as is ever possible in the contingent world of science.
Popular concern about radiation originates with the harmful effects of ionizing radiation, and to a lesser degree ultraviolet, or UV radiation. Ionizing radiation includes subatomic particles and those electromagnetic waves such as X-rays and Gamma rays which are energetic enough to remove electrons from atoms, breaking atomic bonds and creating ions. Ultraviolet radiation consists of electromagnetic radiation with slightly to moderately higher energy levels than those in visible light. It can alter certain bonds in DNA, causing mutations leading to skin cancer in humans and to death in the case of many fungi and microbes. For the latter reason, it is often used in the sterilization of food and of objects where microbial contamination is a concern.
Ionizing radiation is produced in small doses by medical X-rays, CAT scans, and PET scans. Space flight and high altitude flight also expose people to low doses of ionizing radiation. These forms of exposure are systemic, e.g. effecting the whole body, and for essentially everyone except for professional radiologists and pilots the doses involved are too low to be harmful, and much lower than the doses absorbed through exposure to natural radiation sources including the radioactive materials normally present in the human body.
The inhalation or ingestion of radioactive substances, which are used in many scientific fields and were once used to produce glowing paint, is a different matter. These produce very moderate levels of total radiation exposure, but because this exposure can be highly concentrated in a small region of the body the risk of cancer from this exposure can be substantial. For non-scientists in the modern world and far from Chernobyl, the most likely source of significant radiation exposure may be radon gas, which sometimes accumulates in people’s basements through the radioactive decay of natural elements found in the ground. Another possible source of such exposure is former KGB assassins, so it is best to avoid antagonizing the Russian government.
On Earth, the only sources of immediately dangerous doses of systemic radiation are nuclear weapons, the worst possible nuclear meltdowns, and for would-be superheroes or supervillains, direct exposure to the reactor cores, fuel rods, or spent fuel of nuclear reactors. Sadly, in reality such experiences are combinatorially (way beyond astronomically) unlikely to result in super powers. The actual effect of such exposure is known as radiation poisoning or radiation sickness. Its severity varies with the dose of radiation received, which is measured in units known as REMs. The effect ranges from temporary anemia, from a 20-50 REM dose (this is 4-10X legal annual occupational exposure, which is 14X natural background exposure), to nausea, fatigue, temporary immunodeficiency, temporary sterility including stillbirth and a 10% chance of death within 30 days, for 100-200 REM, to hair loss, uncontrollable bleeding in many tissues, sterility, months of convalescence, and a 50% chance of death within 30 days for doses of 300-400 REM, to instant death from nervous system breakdown from 5000+ REM. One nuclear bomb can ruin your whole day.