The Fourteen Fires was a chain of volcanoes located in the Valyrian Peninsula, on the continent of Essos. Five thousand years ago, it was in this location where the Valyrians, at the time a community of shepherds, first discovered dragons.The Fourteen Fires erupted four hundred years ago in a massive cataclysmic event known as the Doom of Valyria, which destroyed the Valyrian civilization and shattered the peninsula. The capital city was destroyed in a single day, and subsequently is often referred to as Old Valyria. The entire region was consumed by the oceans, creating the Smoking Sea.
In the books
In the A Song of Ice and Fire novels, the Fourteen Fires are alternatively known as the Fourteen Flames.
The volcanoes remain quite active and have never entirely stopped erupting since the Doom. However, they are not nearly as active as they were during that massive explosion, which shattered the entire Valyrian peninsula. Some of the volcanoes have sunk beneath the waves, but even those under water continue to belch forth fire and smoke visible on the surface. Sailors fear to even pass within sight of Valyria's ruins, and the glow of the Fourteen Fires can be seen on the distant horizon by passing ships.
Discovery of dragons
The Valyrians first discovered dragons nesting in the warmth of the Fourteen Fires, though no one knows exactly where the species originated. The World of Ice and Fire sourcebook recounts how various cultures throughout the known world have their own legends regarding the origin of dragons: the Valyrians themselves claimed that dragons were born directly from the Fourteen Fires, while in Qarth legend claims that two moons once existed in the sky. When one of them passed too close to the sun, it cracked like an egg and released one million dragons into the world (this story was recounted to Daenerys Targaryen by her handmaiden Doreah in the episode “The Kingsroad”).
Other than Valyria, the only plausible origin site for dragons (at least within the known physical world) is Asshai and the surrounding Shadow Lands: dragon bones and petrified dragon eggs are frequently found there, and the inhabitants claim that dragons existed in the region since before Valyria was founded (they also claim that Asshai is the oldest city in the world, but no one can confirm or deny this). Either way, the people of Asshai never succeeded in taming dragons and using them as beasts of war like the Valyrians did.
Regardless of their origins, it's known that dragons were present throughout the world long before the Valyrians tamed them: Ancient tales say that dragons once flew the skies all over the world - from Westeros to Yi Ti - in the Dawn Age before the Long Night. These tales are supported by the discovery of dragon bones and skeletons in widely- scattered locations, including the island of Ib and in the jungles of Sothoryos, where the Valyrians never had a significant presence.
The volcanic mountains of the Fourteen Fires contained vast mineral wealth; copper, tin, iron, gold, and silver were all found in great abundance. Dragons were used to forge massive quantities of iron into Valyrian steel, which was used to make not only weapons but also gigantic building projects. Slaves were brought in by the hundreds of thousands to labor in the mines, brought in from hundreds of nations across the Freehold and the known world. Life in the mines was dangerous, brutal, and very short, but the Valyrians' demand for ore was insatiable. Some suspect that the disaster of the Doom was actually caused by excessive mining under the volcanoes - possibly even intentionally caused by the slaves using magic as revenge against their masters (though still others say the Valyrians themselves caused the explosion in an attempt to harness some sort of magical power through a spell gone wrong).
The death-worshiping cult of the Faceless Men actually originated in the slave mines under the Fourteen Fires. The thousands of slaves from a hundred different lands faced death on a daily basis, and over time, the belief developed among some of them that while they came from many different lands and followed many different religions, all of these religions were fundamentally the same, because they all worshiped death in some fashion - either polytheistic religions that had a god devoted to death, or monotheistic religions with a deity that had dominion over death. In time this turned into a syncretic belief system worshiping the Many-Faced God of Death: holding that Death has appeared to humans under many different faces as many different gods, but they are all fundamentally one god. They started out by giving the "gift" of death to slaves who were suffering particularly horribly in the mines, to ease their pain - and in time, they learned to surreptitiously give the "gift" to some of their Valyrian masters as well, becoming skilled assassins. Many centuries later, after Valyria was destroyed, the surviving Faceless Men relocated to Braavos, which was founded by ex-slaves who had escaped from Valyria. The Braavosi welcomed them because they had both formed in opposition to the Valyrians, and in Braavos the Faceless Men established a firm headquarters where they continued to operate for the next four centuries.