Dwarves are, of course, another staple fantasy species that have been part of D&D since, if not its inception (as that weird "Elves and Dwarves as classes because we don't have a concept of races" I believe, of Basic D&D) then since the very early days. And of course, they are are also part of the Tolkien tetrad, which is probably why the 5E basic rules provided for Humans, Dwarves, Elves, and Halflings.
Funnily enough, the TvTropes page for Dwarves is "Our Dwarves are All the Same," because there are some assumptions made about Dwarves that are often more rigid than other classic fantasy species. WoW, largely by virtue of a shared origin with Humans, Gnomes, and some other non-playable species (and kind of Orcs, though there's a more tenuous connection to their origin) does make them a bit distinct, but they're not going as far as, say, Draw Steel's dwarves.
Like humans, Dwarves in WoW are descendants of the Titanforged, sentient constructs built by the Titans to transform and safeguard Azeroth (and also kind of shape it to their liking - look, I'm still a "Titans were mostly good" believer, though there's certainly some elements we might call problematic about their approach to Azeroth). The Dwarves were descendants of the Earthen before they were changed by the so-called Curse of Flesh, whose role was to excavate the deep places of Azeroth and construct the extensive, massive machinery there.
Like your classic Dwarves, Azerothian Dwarves have a long tradition of mining, due to their original purpose as excavators. What's interesting is that this prediliction for digging also translated to a grand and proud tradition, at least among the Bronzehammer Clan, of archaeology. It was thanks to Dwarven archaeologists that we players at least first learned of the Titans and their role in creating, well, at least the Dwarves.
But let's talk about the various dwarf clans, and then about the surviving, still-rock-and-metal Earthen, who are also playable. Note that Bronzebeard and Wildhammer Dwarves will likely be represented simply by the standard D&D Dwarf species, while Dark Irons and Earthen will have their own (I think).
Most modern Azerothian Dwarves are denizens of Khaz'modan, an extensive kingdom spanning the middle of the Eastern Kingdoms. Once united under the Anvilmar Kings, who ruled from the mountain city of Ironforge, the death of the last king of this line led to a three-way civil war between the Bronzebeard, Wildhammer, and Dark Iron clans.
Bronzebeard Dwarves:
The Bronzebeards are the largest Dwarf clan, and have held Ironforge since the War of the Three Hammers, though currently they share it with the other major Clans, even if it is their "home territory."
The Bronzebeards have been staunch members of the Alliance since the Second War, and are a highly industrial people, making use of the city's eponymous massive forge and likewise taking advantage of ancient allies, the technologically-advanced Gnomes of Gnomeregan, in their city that sits just across the valley of Dun Morogh from Ironforge, to produce sophisticated machinery at an industrial scale, including airships, tanks, and flying machines.
The Bronzebeards were historically ruled by a hereditary monarchy. King Magni Bronzebeard was the sovereign, and his younger brothers, Brann and Muradin, were also prominent members of the Clan. Brann is probably the most famous and respected adventurous archaeologist on Azeroth, while Muradin was a great warrior who had the unfortunate fate of accompanying Arthas to Northrend. Left for dead after Arthas claimed Frostmourne, Muradin was later discovered to have survived with amnesia and returned to Ironforge years later.
When the dragon Deathwing shattered the world, Magni sought out ancient wisdom to help save the planet, but was transformed into a diamond statue. He eventually regained the ability to move and speak, but his status led to the creation of the Council of Three Hammers, which became the ruling council of Khaz'modan, with representatives of each of the three major clans on it. Muradin currently serves as the Bronzebeard representative.
Wildhammer Dwarves:
Unlike the Bronzebeard and Dark Iron Dwarves, the Wildhammer are drawn to open spaces and fresh air, and have a deep and close bond with their gryphons, whom they ride in the sky as flying warriors, and have long been central to the Alliance's air superiority strategies.
After their exile from Ironforge, the Wildhammer relocated to a similar "interior of a mountain built around a giant forge" city of Grim Batol, to the northeast. However, while they made peace and even allied with the Bronzebeards, the Dark Irons remained hostile to them. The Dark Iron Sorcerer-Queen Modgud brought with her Xal'atath, Blade of the Black Empire, an ancient ritual knife containing the soul of a profoundly powerful void-aligned creature. (Notably, no longer in the dagger, Xal'atath has been the main villain of the current and previous expansion, and when she was still in the dagger, she was the Shadow Priest weapon in the Legion expansion.) While the Dark Irons were turned back, Modgud's dying curse, using the Blade, left Grim Batol tainted with evil and uninhabitable.
While the Wildhammer still have settlements outside of Grim Batol, their de facto capital is Aerie Peak, off to the north in the subcontinent of Lordaeron, not far from the Plaguelands.
The Wildhammer are somewhat more spiritual and tied to the elements and the natural world, and are the main proponents of shamanism among the Dwarves. They are represented on the Council by Falstad Wildhammer, High Thane of the clan.
Dark Iron Dwarves:
The Dark Iron remained hostile to the other dwarf clans long after the War of the Three Hammers. Following the death of his Queen Consort Modgud at Grim Batol, the Dark Iron Sorcerer-Thane Thaurissan sought a power that could turn the tide of the war, seeking out a power he had learned of beneath the Redridge Mountains (the mountains between Khaz'Modan and the human kingdom of Stormwind). The power he found was the Firelord Ragnaros, elemental lord of Fire, and upon summoning him, a massive swath of the Redridge Mountains became a fiery, charred wasteland, and the Dark Iron Dwarves were suffused with elemental fire, leaving them with grey, ashen skin and a core of elemental fire, visible in their eyes and hair.
The Dark Irons built an extensive city throughout Blackrock Mountain, the largest peak in the Redridge Mountains, eventually calling this Shadowforge City, where a cult to Ragnaros became the state religion, and the totalitarian state engaged in slavery and all other manner of terrible acts.
However, the Dark Irons were not all evil. The story, though, gets complicated when we introduce a new element: Moira Bronzebeard.
Moira, the only child of King Magni Bronzebeard, was thought to have been kidnapped by Emperor Dagran Thaurissan, and brave heroes of Azeroth descended through the Blackrock Depths (the lower parts of Shadowforge still firmly in the Dark Irons' control - it's a bit of a distraction, but the Horde actually took Blackrock Mountain as its main stronghold during the First and Second Wars, but the upper parts of the city are still held by a hard-line extremist faction that doesn't recognize the authority of the modern Horde, known as the Dark Horde). Coming to "rescue" Moira, they discovered, after killing the Emperor, that Moira had come of her own free will, and was genuinely in love with the Emperor and also pregnant with his child.
After her father was transformed during the Cataclysm, Moira came to Ironforge to present her infant son as the rightful heir to both the Bronzebeard and Dark Iron clans (guess it's still male primogeniture? Come on, if England can update this, surely the Bronzebeards can). This obviously caused a lot of confusion and panic, but while things looked headed toward some sort of renewed war between the dwarves, the Alliance ultimately intervened and mandated the creation of the Council of Three Hammers, with Moira serving as the representative for the Dark Irons as a kind of Queen-Regent before her son, Dagran, named for his father, would come of age.
Millennia (I think) of strife and conflict have largely resolved, with the Dark Irons now fully integrated into the Alliance (it didn't hurt that Ragnaros was slain, and actually permanently after he was killed within the Firelands) and the three major dwarf clans working together.
Earthen:
The Earthen are the original ancestors of the Dwarves, a Titanforged people created, as mentioned above, to excavate underground and to build and maintain Titan machinery. The Earthen are, true to their name, made of stone and metal, with hair like copper wire and gemstones emerging from their stone skin. They are, though, otherwise largely similar in shape and appearance to Dwarves.
While most Earthen were affected by the Curse of Flesh and became modern dwarves, there are pockets of Earthen to be found in various regions, including the Titan sort of "central command" facility of Ulduar, far up in Northrend, as well as the mostly-underground realm of Khaz'Algar, which houses the Coreway, a passage that leads down toward the Worldcore, the Titan facility that houses the Worldsoul of Azeroth.
The Earthen are sentient, but they are much more construct-like, with a culture built on ironclad laws and responsibilities. That said, over the eons that the Earthen have worked and maintained the facilities, they have deviated somewhat from the programming, developing behaviors that sometimes go against the Titan's "edicts."
Among the traditions of the Earthen is the existence of the Thraegar - Earthen who take on gem-like forms and have immense power. Magni, when he appeared to the Earthen of Khaz'Algar, was mistakenly, or maybe not mistakenly, identified as such. The Thraegar seem to be directly empowered by the Worldsoul, which is what Magni had been attempting to commune with when was transformed.
As constructs, Earthen society is very different - bodies and souls are sort of recycled through Titan machinery, with memories of individual Earthen archived in some kind of magical digital record. As alien as this existence seems, they also have proud cultural traditions, including theater (though their theater also might actually be Titan propaganda, a method of indoctrination that has evolved into an art form).
While they naturally have kinship with the Dwarves, the Earthen are not inherently tied to loyalty with the Alliance, and so there are Earthen who have joined both Alliance and Horde.
The other ones:
So, the following aren't playable, but are worth taking a look at:
Troggs:
It's not clear if this was the result of the Curse of Flesh or some further mutation, but some Earthen transformed into aggressive cave-man like beings known as Troggs. Not to be confused with the reptilian Troggs of D&D, these are actually far more like Grimlocks.
Frost Dwarves:
Apparently some kind of independent offshoot of Dwarves from Northrend, the Frost Dwarves rescued Muradin after the incident with Arthas and nursed him back to health, later elevating him to a position of leadership thanks to his martial skills. That's... like all I got for these guys.
Grummles:
The Grummles are native to Pandaria. The Mogu, themselves a Titanforged species that reverse-engineered the Curse of Flesh to transform themselves back into stone, and generally ruled as brutal tyrants before the other Pandarian peoples rose up against them, peformed a lot of weird experiments on the other peoples of the continent.
When the Mogu got their hands on the Troggs, they performed flesh-shaping magic on them, transforming them into the Grummles. With humanoid intelligence restored to them, they look nothing like the Earthen or Dwarves they once were, and are more rodent-like. Peerless mountaineers, they live on the great mountain of Kun Lai in the northern part of Pandaria, and act as guides and traders, and engage in Yak husbandry.