There are two nuclear reactions that release energy: fission and fusion. The former means splitting of the atom and the latter means the opposite. Fission releases much less energy but fusion takes a lot of effort to do and control.
The way nuclear power plants generate electricity is actually pretty simple: energy released is used to heat up water, which in turn turns to steam and turns a generator turbine.
If there is a failure, radiation could come from the radioactive (unstable) materials used in the process. Or the waste materials which we end up with after the atom is split. There is always radiation coming out of them but they are always under control if everything goes to plan.
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