Out of respect(or fear) of King Anvilmar, the three factions didn't shiv eachother in the streets at every opportunity. This only lasted as long as King Anvilmar did, unfortunately. He eventually died of old age without an heir, and Ironforge Mountain turned into a cage match. Made out of rocks. A rock-cage match. This civil war lasted a few years, but eventually the Bronzebeard Clan came out on top. To add insult to injury, the Bronzebeard threw the other two clans out of Ironforge entirely. This is why we can't have Dwarf Mages.
The Wildhammer traveled through Dun Algaz and founded a brand new kingdom in a brand new mountain, because man, they didn't want that smelly Ironforge anyway. Their kingdom was named Grim Batol. The Dark Irons traveled to what was then part of the Redridge Mountains and founded their new kingdom. Thaurissan, being the humble fellow that he is, named his new city after himself. Thaurissan's city was pretty cool, and the Dark Iron Clan was doing pretty well for itself, but Thaurissan had a few problems on his hands. For one, he still hated the Bronzebeard. For two, he still hated the Wildhammer. For three, his wife's name was Modgud. Yeah. Modgud.
Emperor Thaurissan decided he wanted all of Khaz Modan for himself, so he planned a two-pronged attack on both Grim Batol and Ironforge. Thaurissan and his half of the army would take out the Bronzebeard, and Modgud and her half would take out the Wildhammer. Things were going good for Thaurissan... until he actually got to Ironforge. The Bronzebeard Clan formed up into the world's greatest Flying V and steamrolled the Dark Irons. Thaurissan got absolutely punked. The Dark Irons ran away. Modgud did far better than her husband. Modgud actually reached the interior of the fortress and wreaked some havoc, only to be slain by Thane Wildhammer himself. Her half of the Dark Iron forces also tried to retreat, but reinforcements from the Bronzebeard Clan steamrolled them, too. The War of the Three Hammers came to an end.
The moral of the story is: The Bronzebeards were awesome. Tune in next week when... oh, wait, I'm not done!
The Wildhammer went and lived somewhere else, because Modgud's magic made it all shadowy and scary or something, I'm not sure, but they didn't want to live there anymore. It was tainted. They live in the Hinterlands now snuggling with gryphons. The Bronzebeard are still hanging out in Ironforge, and Thaurissan accidentally blew up his city(and himself) by summoning Ragnaros and created Blackrock Mountain. The other two clan leaders have passed away due to more sane circumstances over the years, and all three clans are now led by their sons.
During the Second War, the Orcish Horde found Grim Batol, thought it was pretty sweet, and took it as their own. This is where the Dragonmaw Clan of Orcs held captive Alexstrasza, Queen of the Red Dragonflight, thanks to the Demon Soul. The Dragonmaw forced the Red Queen to breed and used her children as mounts and warbeasts, which was very bad for everyone that wasn't an orc. Even after the fall of the Horde and the end of the Second War, the Dragonmaw had a firm grip on Grim Batol and the surrounding area thanks to their dragon-making machine Alexstrasza.
Until Deathwing stuck his nose in, that is. You see, Deathwing is crazy. Deathwing tried to wipe out the other Flights and all of the mortal races a number of times, and this essentially put the Black Dragonflight on the brink of extinction. Thus, Deathwing manipulated the Dragonmaw Orcs into bringing Alexstrasza and her eggs out into the open, and he would steal the eggs and raise those dragons as his very own, and proceed to kill Alexstrasza. Luckily, Mary Su-- er, Rhonin and his band of merry men(just one, I guess, and a woman) showed up to save the day thanks to Alexstrasza's lover despite being manipulated by Deathwing, and the Destroyer was chased off. Alexstrasza was freed, the eggs were safe(sort of) and the Dragonmaw Clan was purged by fire and brimstone.
Currently, the Red Dragonflight occupy Grim Batol. What they're doing there is unknown, but it is definitely something important. Rows and rows and rows of draconic warriors surround the fortress, and its heavy gates are locked tight. It is quite possible they're simply defending their Queen, stricken with paranoia and fear after the Dragonmaw dominated them so easily, but there are many other theories as well. One of them being Alexstrasza holds Deathwing captive within Grim Batol, but I find that unlikely, especially considering the revelations in the Burning Crusade.
And that's that for Grim Batol. If you'd like to read further on this topic, there's a few good sources out there. First, you could swing by the Scarlet Monastery or the Hall of Explorers in Ironforge. There's a book in each of those locations about the War of the Three Hammers. For further information on the Dragonmaw Clan's time in Grim Batol, you could pick up a copy of Day of the Dragon. And as always, you can hit up WoW Wiki for additional information, but be warned that it may not be 100% accurate since it is a wiki.