There is some potential danger in operating a nuclear power plant (cf. Chernobyl and Three Mile Island), but the principal disadvantage is the need to safely store the spent fuel for tens of thousands of years.
With proper regulation, safety features, and common sense pretty much none. When you test a poorly designed reactor, lets say a light water graphite moderated reactor also know as an RBMK reactor (Chernobyl) under low power low coolant conditions with al of the safety features turned off things can happen. When you ignore what the computers when they say to shut down the reactor and keep it running (Three mile island) things happen. But if you need disadvantages:
Radioactive Waste
Higher level waste can be reprocessed into Plutonium either through the use of a reprocessing facility or can be prevent from being made by using a breeder reactor where the higher level waste is not made and enriched Plutonium is made instead.
Proliferation of fissionable materials
The problem with breeder reactors is also the reason why they are good.They produce more fuel but they produce more fuel this highly enriched fuel can be used to make nuclear weapons, dirty bombs, or be used in a nuclear reactor to make energy.
There is a large safety risk, from accidents and from terrorists stealing the fuel/byproducts to make weapons of mass destruction, nuclear power plants cost a lot to start, the nuclear fuel is non renewable and there is only enough to last approx 70 years.
Nuclear fuel is in fact renewable in so-called "breeder" plants. France uses them, though there are none in the US.
Nuclear plants are quite expensive to build, but so are coal-powered plants of equivalent capacity. Several companies are working on low-capacity plants... enough to power a neighborhood or a small village... that would be much cheaper.
One disadvantage not mentioned above is the problem of disposing of spent... but still radioactive... fuel.