Twin Dangers
Two major worries around nuclear power are the twin dangers of nuclear proliferation and of terrorism.
Nuclear proliferation refers to the increasing group of nations which possess and/or seek to possess nuclear weapons. Most thermonuclear power plants require enriched uranium fuel. If you have facilities to enrich uranium for power generation, the same technology can be used to enrich it further into “weapons-grade” uranium. Weapons-grade uranium has a high enough 235U concentration to form a “critical mass” in a small space. In general, weapons-grade uranium has over 80% 235U. At 85% 235U, a “critical mass” capable of exploding is 85kg (with a density of 19 kg/litre, this represents a volume of just 4.5 litres). As the uranium is enriched in 235U, the critical mass decreases (critical mass of 100% 235U is ~50kg), making the resulting nuclear weapon smaller, but equally powerful.
Another problem is that all thermonuclear reactors create plutonium. Because plutonium is a different element, it can be separated from uranium using chemistry. This is far easier and cheaper than to separate 235U from 238U!
Nations which have a reactor may decide to create a plutonium weapon instead of enriched uranium. Using pure plutonium (239Pu), critical mass is 10kg. With a density of 20 kg per litre, this is ~0.5litres or the size of a grapefruit.
The volume of uranium (235U), or particularly plutonium (239Pu), required to make a weapon is very small and easily hidden. This presents the worry that terrorists could acquire the material and create a crude nuclear weapon with which to threaten people. For various technical reasons, this would probably not be capable of a powerful explosion, but could be “dirty” spreading radioactive material far and wide.
Worry increased substantially when the Soviet Union (USSR) collapsed. There was a rapid increase in Russian lawlessness, and a cut in Russian military spending. The latter caused many people who had been employed making, testing and guarding nuclear weapons to lose their jobs, as well as scientists attached to the military. Some people are thought to have sold fissile materials to terrorists. Indeed some stolen uranium & plutonium has been recovered. American stockpiles of uranium and plutonium are also a worry. Regular inventories are required, but some of the figures do not add up. Were materials lost, stolen, or sold, or were these just accounting errors? Due to government secrecy, these questions have not been answered publicly.