In terms of fission, a nuclear chain reaction is a continuous, sustained splitting of atoms. Materials capable of sustaining such a reaction include U-235, Pu-239, Th-232, and some others.
Basically, you fire a stream of neutrons into one of those materials. The neutrons collide with individual atoms, splitting them and releasing more neutrons. If the material is dense enough, those neutrons go on to split more atoms, and so on. Thus, the reaction sustains itself.
The faster a neutron is travelling, the less likely it will hit an atom, thus most nuclear chain reactions take place in a hydrogen-rich environment, as hydrogen has a slowing effect on neutrons.
Reactions that are intended to have a rapid, devastating effect require little in the way of control. See atomic bomb. Otherwise, the chain reaction can be controlled by way of addition or removal of cadmium, which absorbs neutrons. The control rods present in all nuclear reactors are made of cadmium.

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