The reaction that takes place in nuclear power plants and nuclear bombs is nuclear fission, which is a chain reaction. Fission is the splitting of atoms. Nuclear reactions are different from chemical reactions because they involve the nucleus, while chemical reactions only involve electrons outside the nucleus.
In a nuclear reactor the chain reaction is controlled and sustained for the purpose of producing heat, which is tranferred to water which turns to steam and moves turbines. Some reactors produce medical isotopes or plutonium for weapons.
Fissionable materials such as uranium-235 have unstable nuclei. When they absorb neutrons they undergo radioactive decay to produce heat, radiation and addtional neutrons that trigger fission of more atoms. The nucleus is split into lighter nucei.
The rate of the fission is controlled in reactors using moderators, which are substances that absorb the neutrons being produced to prevent them from triggering fission. Moderators aren't used in nuclear weapons, which explains the explosive nature and tremendous amount of energy released at once vs. the more steady nature of nuclear reactors.