In Star Trek, Dilithium is a crystalline substance used mediates the reaction between the matter and anti-matter in a starship's main engine core and dilithium crystals control the powerful matter/antimatter reaction which permits the ship to travel faster than light.

There does exist in real science a substance called dilithium - Li2 is a diatomic molecule comprising two lithium atoms covalently bonded together. Lithium in its pure form does not occur naturally on  Earth. It is a soft, silver white metal - the lightest of all metals, in fact. It is used primarily in heat-transfer applications, batteries (mainly cell phone and camera batteries), household  appliances such as toasters and microwaves, and in high performance alloys such  as those used for aircraft construction. Lithium is one of only four elements theorized to have been created in the first three  minutes of the universe through a process called Big Bang nucleosynthesis. While lithium is a metal, dilithium is only known in the gas phase. It has been observed, for example, that 1% of lithium in the vapor phase (by mass) is in the form of dilithium.
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how it works

Just as the fictional dilithium in Star Trek was an important source of fuel, lithium deuteride (compound of lithium and hydrogen, LIH) was the fusion fuel of choice in early versions of the hydrogen bomb. When bombarded by neutrons, both Li6 and Li7 produce tritium. Tritium fuses with deuterium in a fusion reaction that is relatively easy to achieve. Although details remain secret, lithium apparently no longer plays a role in modern nuclear weapons, having been replaced entirely for the purpose by elemental tritium, which is lighter and easier to handle than lithium salts. Lithium will be used to produce tritium in magnetically confined nuclear fusion reactors using deuterium and tritium as the fuel. Tritium does not occur naturally and will be produced by surrounding the reacting plasma with a 'blanket' containing lithium where neutrons from the deuterium-tritium reaction in the plasma will react with the lithium to produce more tritium. Various means of doing this will be tested at the experimentasl ITER fusion reactor being built at Cadarache, France.



Dilithium, as depicted in the Star Trek computer
game Elite Force