In Final Fantasy VI, Magitek acts as the general term for technology powered by magic forcibly extracted from espers by the Gestahlian Empire, while in Final Fantasy XIV, Magitek refers to the Garlean Empire's ceruleum-fueled devices.
Magitek was invented by Cid Del Norte Marquez, an engineer for the Empire. Using an unknown process, Cid discovered how to forcibly extract magic from espers and imbue machines and humans with magical energy. As the first experiment with Magitek, Cid infused Kefka Palazzo with magic, giving him great power but destroying his sanity.
Later experiments used a modified process, granting the subject magic but at a lower potency. This increased the usefulness of the process, and the Empire began to create many machines that could use Magitek power. This lead to the construction of the Magitek Research Facility in the Imperial capital Vector, where espers were drained of their powers and machines and humans were granted magical abilities.
The main use of Magitek is the Empire's signature Magitek Armor, which can fire beams of magical energy. Magitek Knights, humans given the ability to use magic, also begin to appear in the Empire's ranks with increasing frequency as the Magitek processes are developed. The animals around the continent of Vector also became imbued with magic, possibly from the fallout from the Magitek Research Facility. With their armies using Magitek technology, the Empire conquered the southern continent and attempted to push into the north, successfully conquering Doma and Figaro.
After the Returners attack Vector, the Magitek Research Facility is heavily damaged and the espers used to power the Empire's experiments turn themselves into magicite and are taken, effectively cutting off the Empire's supply of Magitek power. Cid and Kefka discover from the Returners that the magic taught by magicite is more potent than magic extracted through Magitek. Thus, the factory is dismantled and, with the war approaching its end, Magitek is largely discontinued in use as Kefka and Gestahl opt to use magicite instead.
Following the end of the world, the Empire is destroyed, but Kefka's Tower still contains a great deal of machinery and Magitek soldiers, only they are more powerful than the Empire's, likely due to Kefka's increased magical powers and ability to imbue others with magic.
Magitek refers to machina devices that are fueled by ceruleum, a rare substance created by draining the energy of crystals. On Hydaelyn, the Garlean Empire are the foremost users of magitek. For reasons currently unknown, the majority of Garlean citizens have a natural ineptitude for the workings of magic, and so to overcome this disadvantage developed magitek.
The technology is not widely used in the Eorzean region, but is slowly being adopted and seeing more widespread usage. It will not be long before a merging of Garlean magitek and Ul'dahn mine carts results in a web of iron causeways spread across Thanalan. Magitek Armor first saw military use during the great battle of Carteneau.
While some Airships are fueled by ceruleum, they are not grouped under the label of magitek.
"Magitek" is a portmanteau of "magic" and "technology."