So, in my short time here I've learned a lot about the Arcanist class. I'm by no means an expert, but I've looked into several different sources to make sure I had the basics for the Arcanist down. At the end of the day this is pretty much just for me to look back on without going to the several different sources to make sure I have everything.
Where it All Started: Arcanima was originally practiced in the South Sea Isles, and five hundred years ago or so was brought to Limsa Lominsa where they were greeted as letter men. Over the years they lost more and more knowledge of the school of arcanima, and to prevent further loss created the Arcanist’s Guild.
Arcane Geometries: this is the main tool of the Arcanist and is accompanied by their Grimoires. A common theme with all casting classes (with the exception of Red Mage) is the fact that they draw upon the Aether to use their magic. Arcanists are no different, but instead of directly taking from the Aether they use geometric symbols and precise calculations to change the Aether into vastly different spells. This results in their spells like Ruin, Bio, Miasma, and Shadow Flare.
The Grimoire: this is all pretty much speculation. One source says that it's the Grimoire itself that provides the Arcanist with their knowledge of geometries and arcanima, but the lore and the way the Grimoire is presented in game looks more like a combination of a text book and a note book. I have to emphasize that this entire next portion is entirely speculation. It could be argued that the geometric patterns written in the book are written by you (that's what I'd say if your character is skilled enough in the art), as it seems the pre-written sigils are enough to create the effects of the spell. With spells like Miasma and Ruin we see the Arcanist hold their Grimoire out in front of them while their hand moves behind their shoulder. What's likely happening is they are using their hand to draw in Aether from the area and then using the geometric patterns within the book to manipulate the Aether into the desired effect. Ruin II and Bio are likely very simple spells that are able to just be whipped out from the pages due to how little Aether they both require. This makes sense for Ruin II due to the fact that it is a spell learned later, after we have begun mastering the original spell of Ruin. Ruin III is where things get mixed fancy, BUT Ruin III isn't technically an Arcanist spell. It's a Summoner Spell. Nevertheless, I'm not a stingy person and I'd argue a skilled enough Arcanist can cast Ruin III. With Ruin III the Arcanist holds their Grimoire out in front of them while moving their hand. The hand movements are vastly different that those used when casting Ruin or Miasma because in this case their fingers and wrist is actively moving to create a pattern. It's likely that the Arcanist in this stage is doing the following: they're using the geometric pattern of Ruin within the Grimoire, while taking in Aether within the area, while adding to the effect by tracing out their own Geometric pattern on the fly to further enhance the Ruin spell. When the spell is complete the Arcanist shifts their book to be directly in front of them and at the same time claps down their hand on the page, thus completing the casting of the spell and applying the heightened geometric symbol to the book to fully enhance Ruin. I'd talk about Ruin IV and other spells, but honestly I haven't actually gotten that far in the Summon/Arcanist class, but I'll add those to this once I'm there. I assume the principles are the same, assuming they even change the casting animation.
Carbuncles: now these are some controversial little cuties. It's heavily argued among the forums whether or not a Carbuncle maintains their memories and personality each summoning, and I'd argue they do. To create a Carbuncle an Arcanist must have a gem that corresponds to the Carbuncle's attribute. Emerald Carbuncles are aligned to wind, Topaz Carbuncles are aligned to earth, and Ruby Carbuncles are aligned to fire. I'm not sure about the other ones because we don't really see them that often except when used by Alphinaud. Creating a Carbuncle relies on manipulating the Aether that is within the gem and the Aether in the area with geometric patterns. Because of the reliance on the gem, I'd argue that the gem itself acts like a memory card and the manipulation of the Aether provides the Aether within the gem a body. So, you're effectively summoning the soul of a Carbuncle through the gem, and creating a body for the soul through the Aether around you. Thus, your Carbuncle has the same memories and personality of your past summoned Carbuncles (because it by all rights is the Carbuncle you've been summoning in the past), but just has a different body to occupy.
How People React to Carbuncles: I imagine most people (especially those in Limsa Lominsa) would see a Carbuncle as nothing more than a tool of the Arcanist or a cute little critter (it's hard to be cold to something that's adorable) so, unlike a Warlock in say World of Warcraft, a Carbuncle would be seen with fondness, as a spectacle, or as a tool. No one would attack an Arcanist if they had their Carbuncle out. To an Arcanist on the other hand that would vary greatly from person to person and is pretty much up to them on how they'd see them. Arcanists are well acquainted with Carbuncles and as such likely either see the Carbuncle as a companion, a friend, a pet, or even just a mindless tool.
Playing God: The nice thing about summoning Carbuncles through specific and personal geometric patterns is the fact that you can technically alter your Carbuncle's personality to your own whim. I imagine it's not all that complicated when summoning a Carbuncle to alter specific values to give it certain quirks, personality traits, or flaws. You could then in theory have a Carbuncle that is completely aloof and effectively a robot, or you could have a Carbuncle that's full of life and energy and just loves getting attention. You could have one be friendly and loyal, or you could have one be aggressive and act more like a guard dog. The sky is the limit.
Speculation with the Quill: I believe that the quill that the Arcanist uses is just as important as the grimoire itself. In game play we see the Arcanist write into the grimoire with a quill (with no ink provided, though that might just be me being critical), lifting the geometric shape that they create off the page and literally flashing it before our eyes to create their Carbuncle. Note that they don't do the same with any other spells which is why I think the Grimoire is a cross between a note book and a text book. It could be argued that the quill is augmented with Aether, similar to a magitek device, but far more basic in nature and not quite on the level of being called magitek. In theory this would provide the user with the ability to trace out a non-permanent version of a geometric pattern, and give them the ability to lift it off the page to provide the geometric pattern the ability to manipulate the Aether within the gem or around the gem to create the Carbuncle.
Speculation is Key: I'll point out again that a lot of this guide is 100% theory and doesn't have much basis in lore and is simply from my observation. The way Arcanists, Carbuncles, and Grimoires work is only loosely covered in the lore, and honestly I think Square did this on purpose to leave it all open to their RPers. The plus side of being an Arcanist is that a lot of what you can do is very flexible since it all relies on mathematical equations and formula, so by altering certain variables you can get vastly different outcomes. It's likely you could create a far more offensive based Arcanist by altering the way you write down your Arcane Geometries, or be far more supportive and more of a healer by doing the same. As a final note I hope you enjoyed this read. I'll link my sources below.
Edit: did some cleaning up of the entry to make it a lot more organized now that I have the time to do so.
https://ffxiv.consolegameswiki.com/wiki/Arcanisthttp://finalfantasy.wikia.com/wiki/Arcanist_(Final_Fantasy_XIV)
Encyclopedia Eorzea pages 8, 104, 234, 301