The Sea of Clouds can refer to two things in Final Fantasy XIV. Firstly, it refers to the whole Sea of Clouds area above and among Abalathia's Spine. Secondly, it is the in-game named used to refer to certain regions within the Sea of Clouds itself. In Japanese, these regions are known as 'Abalathian Sea of Clouds' and 'Dravanian Sea of Clouds', although the English translation gives them the names 'Sea of Clouds' (with no embellishments) and 'Churning Mists', respectively.
The Sea of Clouds
The Sea of Clouds (雲海 [unkai] or 'cloudsea' in Japanese) spreads out all throughout the peaks of Abalathia's Spine, as far as can be told. It reaches down into the highlands of Coerthas and Dravania, and an unknown distance up into the skies above Abalathia. Since both Coerthas and Dravania can have quite clear skies, it can be deduced that the Sea of Clouds doesn't cover all of the area above them, just parts. It is unknown how far to the north towards Farreach and how far to the west towards Xelphatol it covers. What is known that in certain parts of the Sea, there are numerous floating islands in the sky, capable of bearing life.
The Sea of Clouds is home to specialized varieties of birds, fish and other living things that more commonly live in water rather than the skies. Among the floating islands of the area, various floating types of flora can also be found. As exploration of the Sea has only barely begun due to recent advancements in airship technology, it is unknown just how different things can be deep in the mists.
The Sea of Clouds includes regions like the eponymous Sea of Clouds (Abalathian Sea of Clouds), the Churning Mists (Dravanian Sea of Clouds), Azys Lla, the Diadem, and the airship exploration region.
Abalathian Sea of Clouds
The Sea of Clouds region (アバラシア雲海 [abalathia unkai] or 'abalathian sea of clouds' in Japanese) forms a portion of the Sea of Clouds above Abalathia's Spine. There are numerous floating islands in the area, held in the sky by the overabundance of wind-aspected crystals. Recent advancements have seen an Ishgardian outpost to the area, although people of the five races are still an uncommon sight in the region. The main inhabitants are the beastman race of Vanu Vanu.
The main locations of the Sea of Clouds region are Cloudtop, Last Step, the Wisent Herd, Voor Sian Siran, Vundu Ok' Bendu, Ok' Vundu Vana, Ok' Vundu Mok, the Blue Window and the Gauntlet. There are also the settlements of Camp Cloudtop, Ok' Gundu and Ok' Zundu, the latter two of which are Vanu Vanu villages. Furthermore, somewhere in the general vicinity of the Abalathian Sea of Clouds floats the inert Void Ark.
Churning Mists
The Churning Mists region (ドラヴァニア雲海 [dravania unkai] or 'dravanian sea of clouds' in Japanese) is the second main portion of the Sea of Clouds as a whole. The islands here are in somewhat bigger chunks than those in the Abalathian Sea of Clouds, and are home to moogles as well as dragons - and more specifically, it is the domain of Midgarsormr. There are hints scattered all over that the place used to be a common home for both elezen and dragons during the several hundred years before the formation of Ishgard, but those buildings are now in ruin.
In the game, Churning Mists is accessed from Mourn in Dravanian Forelands via the mountain Sohm Al. It consists of the main areas of Sohm Al Summit, Eil Tohm, Landlord Colony, Four Arms, Greensward, and Ohl Tahn. Somewhere nearby are also the floating isles of Aery where Nidhogg lives, and Sohr Khai where Ratatoskr used to live.
Airship Exploration
There is a group of 22 unnamed islands in a further unspecified region of the Sea of Clouds, that adventurers explore with unmanned (save for mammets) airship voyages. These islands are rich with resources that the mammets bring back with them when they return. They are not reachable by the players themselves.
You can read further about airship voyages here.