The Mechanics of Spell Casting

[Enkidoh]

Well, the FFXIV website basically mentions how spells are generated in the description of each Disciple of Magic, found here:

Conjurer

Conjury calls upon the elements of earth, wind, and water and concentrates them to a potency at which spells can be weaved. Through practiced meditation on the essences of creation, conjurers draw forth and absorb aether from their immediate surroundings. A wand or cane made from unworked wood is then utilized to focus the aether until it manifests as the desired spell. Versed also in magicks that restore and strengthen, conjurers are regarded as accomplished healers.

Thaumaturge

In the hands of a skilled practitioner, thaumaturgy can be a force of terrifying destruction. At the heart of this school of magic lies the ability to call forth and command the latent aether within oneself through deep introspection.
To then mold that aether into sorcery, the thaumaturge makes use of a scepter or staff, within which is housed a medium—a natural stone imbued with magical properties. Thus armed, the thaumaturge is capable of wreaking considerable havoc via ruinous spells and curses.

Arcanist

Adepts of the art of arcanum derive their might from symbols of power born of geometric techniques hailing from across the southern seas. Held within occult grimoires, these symbols lend shape to the arcanist's aether, thereby allowing him to produce myriad powerful spells.
Using the selfsame symbols to unlock the latent power contained within gemstones, arcanists are also able to summon forth the familiar known as Carbuncle to carry out their bidding.

So from that, basically it can be seen that:
CNJs use a wooden staff to borrow aether from the surrounding environment and to focus it into a spell
THMs use the aether from their own bodies and channels it through their bone or metal staff into a spell
ACNs use obscure mathematics to manipulate aether into whatever form or shape they desire (specifically into the metaphysical entity Carbuncle).

Hopefully that hasn't been too confusing.