Garlean magitek run on a substance known as ceruleum. It is aetherial in origin, but doesn't seem to be magickal in any way. It's sort of like the crude oil of aetherial matter; it's mined out of the ground and refined. It's pretty rare in Eorzea (Northern Thanalan is the only large deposit we know of), but on the Garlean lands in the north, it's pretty common. The FFWiki claims that ceruleum can also be made by draining crystals, but I've yet to see this actually sourced anywhere.
The crystal engine was an Allagan invention, and is the main difference between the magick+technology of Allag and the Magitek of Garlemald. The Garlean versions are a non-magickal knockoff.
This one requires a bit more background. Having faced primals in Othard, and knowing of their kind from Allagan records (hence how they came across the term eikon, I imagine), the Garleans do seek to eliminate primals in order to save the world. However, they also feel that annihilating groups, cultures, or even entire regions is completely permissible if it leads to the destruction of primals, which threaten everyone and everything.
Garlemald tried to take Eorzea traditionally, starting with Ala Mhigo in 1557, but the 1562 operation to take Silvertear Lake and deprive the beast tribes of a means to summon primals was so great a failure that primals began to appear immediately (read: I'm not saying it was Ascians, but it was Ascians). There's even some hint that the move on Silvertear was prompted by a primal showing up between 1557 and 1562, but this has never been proven and SE has played very, very coy about it. The identity of the primal in the references has never been revealed. But I digress.
From 1562 to 1572, the Garleans sat on their hands in Ala Mhigo, with no idea how to solve the primal problem. Nael van Darnus suggested just dropping a moon on Eorzea and being done with it. The Emperor, growing old and having failed to take Eorzea thus far, went along with it because he was desperate and because Nael could make it happen. Gaius thought it was depraved, though - he wanted to "save" Eorzea by bringing it under the yoke of Garlean rule by conquest (which only really became possible after the Ascians gifted him the Ultima weapon so he'd stop twiddling his thumbs and make some chaos). Whether that's salvation or not is up for debate, but the Ala Mhigans don't seem to have benefited much from it, what with all the massacres and conscription.