There are fission and fusion reactors. However, at present (2016) there is no commercial fusion reactor which can produce more energy than is required to operate it.
There are two primary design styles of fission reactors to produce electricity. Pressurized, and Boiling water reactors.
Uranium, plutonium or thorium (for fission reactors, by far the most common type).
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Total of 104 reactors, most sites have two reactors
Most plants have two reactors but some have more
Current nuclear reactors rely on nuclear fission as their nuclear reaction.
Most are lightwater moderated and cooled, these are the PWR and BWR. There also a substantial number of heavy water reactors, based on CANDU, and gas cooled reactors mainly now in the UK.
Well, as nuclear reactors are nuclear reactors, nuclear reactors are not used inside nuclear reactors.
There are three nuclear power plants in Michigan. Enrico Fermi, Donald C. Cook and Palisades have one, two and one operating reactors, respectively. As regards active nuclear reactors used for generating electric power, that's the list. The count is one plus two plus one, or four reactors.
The fuel in current reactors is all uranium. Usually enriched to 3% uranium-235.Someday part of the fuel might be plutonium, either recycled from bombs or made in breeder reactors. But not now.
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Graphite is a pure form of coal or carbon. It is a good conductor of heat and electrcity. It is used as a neutron moderator in nuclear reactors of type Gas Cooled reactors.