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It might not seem obvious, but the "original use" of the element Americium was as a tool of scientific investigation. Americium is a synthetic element, so it was made in a nuclear reactor facility by exposing plutonium to the neutron flux in the operating core. It was produced in small quantities to investigate its nuclear, chemical and physical properties. That's what it was originally used for. You can read about it here: http://en.wikipedia.org/wiki/Americium
A breeder reactor is one type of nuclear reactor, but not a type that is in general commercial use at the present time
the nuclear reactor exploded
It could be argued that the use of nuclear power began in the 1950's when a nuclear reactor in the Idaho desert supplied electric power to a local community.
That released by fission in a nuclear reactor
To produce electricity
No. LLNL even tested several Uranium-Hydride bombs in the 1950s. Even though their computer models said the devices should explode, none gave a nuclear yield. One could use the waste from the reactor as a Radiological Weapon, but the reactor itself is not useful as a weapon.
A nuclear reactor will do what is asked here. Use the link below to the related question about what a nuclear reactor is.
If you are a smart guy, you can make a nuclear bomb, or a nuclear reactor...
Heat is eliminated through use of reactor coolant passing over the nuclear fuel in the primary coolant loops.
not practical
A nuclear reactor uses either nuclear fission or nuclear fusion to generate electricity, while bio-reactors use the excretions of many animals to generate electricity.