Along with crystals, elemental magic is a long standing part of Final Fantasy lore. Aside from providing familiar and consistent spell names like Aero, Firaga and Meteor these elements have been deeply embedded in the MMOs created by Square Enix. Well, one of them at least...
In Final Fantasy XI, the elements were involved in almost everything. There were eight elements: fire, ice, water, earth, wind and lightning/thunder plus light and dark. Of course these were associated with your usual Black Mage nukes like Blizzard and Stone but also Ninjutsu spells and a Summoner Avatars. What's more there were eight days of the week, one for each element and a point on the compass dedicated to each element. There was also weather associated with each element, which could result in a storm, also known as "double weather". Most obvious and, I guess, frequently encountered was the double ice weather that caused blizzards but the double light storms in Qufim Island were rare and beautiful.
Each of the elements also had corresponding elemental strengths and weaknesses as shown below:
It was widely believed, but never confirmed, that this elemental relationship even affected synthesis. In other words, attempting synthesis with a Wind Crystal on Earthsday was more likely to result in success, while using it on Iceday increased the chance of an HQ result. The speculation extended to which compass point you faced!
The weather also affected some weaponskills and activated certain bonuses on some equipment. One of the most sought after was the Desert Boots, which made you run faster in a duststorm. When the Scholar job, which could create it's on own localised duststorm, was introduced some people got very rich...
In many ways, the degree to which the elements were involved got a bit daft. If you were a mage and wanted to maximise your damage you needed a corresponding staff and even a belt! The speculation about how the elements affected crafting bordered on tinfoil hat crazy and let's not even get into how the moon affected skill gain!
However, I can't say I'm not disappointed to see a lot of this missing from Final Fantasy XIV. One of the things I loved about Black Mage in FFXI was that you were a true master of all the elements. In FFXIV the elemental spells have been split between Thaumaturge and Conjurer. This does make a lot of sense when you consider that a max level BLM in FFXI would only ever cast Blizzard and Thunder spells as, for some reason, they were created to be the most powerful. Also, the Light and Dark elements have been replaced by the Astral and Umbral dichotomy.
In FFXIV a lot centres on the Twelve, and I expect the days of the week are named for them but if they are, I haven't been able to confirm it. The Twelve also don't seem to be mentioned much in the story, except by swearing pirates. Maybe I'm too new to the game to fully appreciate the implications of the new "affinities" but I suspect that a lot of the details were lost, or never fleshed out, when the original FFXIV died. If that's the case it's totally understandable but the elemental system in FFXI really added a richness to the lore that I am, so far, missing in FFXIV.