The "Sons of the Dragon" era is the period between the War of Conquest, and Aegon I's subsequent rule until his death in 37 AL, and the ascension to the throne of King Jaehaerys I in 48 AL.
Aegon I Targaryen (sometimes known as "The Dragon") had two sons, one by each of his sister-wives: Aenys I, son of Rhaenys, and Maegor the Cruel, son of Visenya. This era was dominated by a long string of anti-Targaryen rebellions, hoping to overthrow their recent conquest. These culminated in the Faith Militant uprising, the largest of these rebellions, which dragged on for many years and was the dominant military conflict of this historical era.
King Aenys I (r.37-42 AL)
Aegon I was survived by his sister-wife Queen Visenya, her son Maegor, and his bastard half-brother Orys Baratheon. His other sister-wife Rhaenys had predeceased him, dying in Dorne, but she already had multiple children with Aegon: their son Aenys and two or three daughters. Being the older brother, Aenys succeeded as king upon Aegon's death - to the consternation of dowager-Queen Visenya and Maegor.
When the two brothers were young, Aenys bonded with the dragon Quicksilver, one of the progeny of Aegon I's original three dragons. Maegor, however, refused to bond with any dragon, claiming none were worthy - or perhaps none of the available ones at the time. Instead, Maegor patiently waited until his father died, and then bonded with Balerion the Black Dread himself, Aegon I's own dragon. Maegor would later use Balerion to devastating effect during the Faith Militant uprisings. As for Aenys, he had always been sickly and something of a weakling in his childhood, though bonding with Quicksilver gave him some new strength and confidence as he grew older.
Still, Aenys recognized that he was not a warrior as his father was. In contrast, Meagor was a preternaturally skilled swordsman (not unlike the later Jaime Lannister), at the age of barely 12 he easily defeated squires five years older than himself. He was one of the youngest men ever knighted, at only 16 years old. When Aenys succeeded to the Iron Throne upon their father's death, he presented Maegor with Aegon I's Valyrian steel sword Blackfyre, admitting that Maegor was always much more of a warrior than he ever was.
While Aenys I wasn't the completely inept weakling that later histories sometimes remember him as being, he still wasn't a very effectual ruler. Overall, he had an eager-to-please personality, desperately trying to win the friendship and loyalty of hostile lords when objectively, use of the sword to violently restore order would have been the more pragmatic and decisive action.
Numerous minor rebellions began popping up early in Aenys I's reign, some of them legitimately balking at the Targaryen's incestuous parentage but many also using this as a simple excuse for trying to overthrow their still relatively new Targaryen overlords. One major rebellion sprung up at Harrenhal, under a man calling himself "Harren the Red" who claimed to be the heir of Harren the Black (apparently a false claim). Another major rebellion sprang up at the Eyrie. Orys Baratheon, now an old man, had served as Hand of the King to Aegon I throughout his reign, and continued as Hand under his half-nephew Aenys I (thus holding the position for over forty years). Orys attempted to handle the local rebellion at the Eyrie but was killed in it. He was succeeded as Hand of the King by Aenys's half-brother Maegor, who promptly crushed the rebellion at the Eyrie, and avenged Orys by throwing rebel Vale lords out the Moon Door there.
As Hand of the King, Maegor quickly established himself as the real power behind the throne. Given that Maegor possessed the Targaryen ancestral family sword Blackfyre, and rode Aegon I's own dragon Balerion, many recognized that he held the real symbols of power.
The exact sequence of events isn't clear, but Aenys made it a point not to continue the Targaryen practice of incestuous marriage, and instead married outside of the family for political reasons - possibly with House Velaryon, one of the Targaryen's followers from Valyria who now ruled Driftmark. In subsequent generations, if there was no daughter born to a current Targaryen generation, they would frequently intermarry with the Velaryons. Particularly, after the first union this already made the Velaryons their cousins, and subsequent unions online increased this incestuous interbreeding - marrying a second cousin from House Velaryon was seen as a second best if there were only sons in the current Targaryen generation. Regardless, Aenys did not enter into an incestuous marriage: he could not go back in time and change his own parentage, but he could at least stop exacerbating tensions with the Faith of the Seven by continuing the practice. Maegor, in contrast, brashly embraced the ideology that Targaryens were above other men, and incestuously married one of Aenys's sisters (Maegor's own half-sister). Though subsequently the marriage was childless. Aenys I's own marriage produced three children: Princess Rhaena, Princess Jaehaerys, and Princess Alysanne.
The Faith Militant uprising
In the first few years of his reign, many localized rebellions sprang up against the Sons of the Dragon: Aegon the Conqueror was a living legend, seen as practically a demigod on the battlefield as he rode astride his dragon. His sons, on the other hand, did not have the same mythical aura of invincibility about them - and if they wanted it, they would have to earn it. Therefore after Aegon died, the recently defeated Kingdoms thought they might have a chance to overthrow the still newly-established Targaryen rule. The first of these were unorganized local rebellions, such as at Harrenhal, or the particularly bad one at the Eyrie which claimed the life of Hand of the King Orys Baratheon. Each of these was crushed in turn, but it became a game of whack-a-mole as no sooner did the Targaryens focus their attention on restoring control over one area, than new rebellions sprang up somewhere else.
Gradually, this growing anti-Targaryen sentiment coalesced around the Faith of the Seven. On ideological grounds, the Faith was disgusted with the Targaryens for practicing incest, in defiance of the Faith's basic moral teachings. Aegon I himself had tread lightly with the Faith, and was strong enough that their leadership did not seriously consider challenging him. Aenys, despite doing his best to placate the Faith by not engaging in an incestuous marriage, could not shake off his own parentage. Seeing that Aenys's rule was weak, the Faith of the Seven finally turned on the Targaryens, and the High Septon declared them abominations born of incest who had no right to rule. Some joined these rebellions out of genuine moral disgust at Targaryen incest, and out of religious devotion to the Faith. Many more, of course, were localized rebellions that simply disliked Targaryen overlordship and wanted to reassert independence, but which joined the swelling tide channeled by the Faith.
For thousands of years, the Faith of the Seven maintained its own military orders, known as the Faith Militant. These consisted of two major orders: the Warrior's Sons were composed of trained knights, younger sons of noble Houses (named after the Warrior aspect of the Seven-in-One God); and the Poor Fellows, composed of simple commoners. The Warrior's Sons were very well equipped and trained for warfare, but not very numerous: the largest concentration ever in once place was the 700-strong chapter at King's Landing. The Poor Fellows, in contrast, were very numerous and could instantly raise new members: any poor farmer who suddenly felt great religious zeal could technically join the Poor Fellows. The result was that in times of open war the Poor Fellows could quickly swell their ranks by basically just deputizing large mobs of peasants. The downside of this, of course, was that they were just mobs of peasants: with no combat training, no defensive armor or horses, and typically armed with only sharpened farming tools. This did mean, however, that the Poor Fellows could replenish their ranks much more easily than the Warrior's Sons could. The military orders which made up the Faith Militant formed the backbone of the new series of rebellions, their ranks swelled by rebellious secular lords who wanted to drive out the Targaryens.
The Faith Militant uprising began when an assassination attempt was made on Maegor (possibly through poison), though he later recovered. Queen Visenya was still alive at the time, and dismissed her son's maesters for the poison specialist and healer Tyanna of Pentos. As soon as Maegor had recovered enough, he appeared outside the Red Keep, and crowds erupted in wild cheers and celebration...which quickly died down as Maegor directly proceeded to mount Balerion the dragon. Maegor then flew the short distance across the city to the fortified sept of the Warrior's Sons, where their 700 members were holding their morning prayers. Without giving any warning or ultimatum, or pausing to determine exact evidence over just how directly involved the Warrior's Sons were in the assassination attempt on his life, Maegor burned all 700 men to death. Dragonflame completely consumed their sept and all within, and the stench lingered around the capital city for days. As events rapidly spiraled into open war, the High Septon responded to the massacre by proclaiming that "all true and pious children of the gods" must take up arms against the reign of "dragons and monsters and abominations".
At a stroke, Maegor had decisively destroyed the largest chapter of the Warrior's Sons - no subsequent unit of the organization, during the rest of the uprising, ever achieved a number as high as 700 members again. Nonetheless, surviving major units of the Warrior's Sons survived in the other major cities of Westeros: Oldtown, Lannisport, and Gulltown (House Manderly of White Harbor in the North does worship the Seven, but apparently White Harbor still has enough of a mixed population that the Warrior's Sons never achieved a great concentration there). Besides these three major cities, another major chapter was also located at the large town of Stoney Sept in the Riverlands. Meanwhile, large numbers of Poor Fellows could be quickly raised in revolt.
The opening advance came from the Reach, proceeding from the High Septon's seat at Oldtown. An army of 9,000 poorly armed commoners who had joined the Faith Militant's Poor Fellows advanced up the Roseroad towards King's Landing. The host was in the middle of crossing the Mander River at the town of Stonebridge when it was attacked by two lordly armies of armored knights on heavy horse. The Poor Fellows, in contrast, were truly just poor peasants: at best armed with wood axes and sharpened farm tools, and wearing only boiled leather. Moreover, they were completely outmaneuvered and in the worst tactical arrangement imaginable. With half of their forces on either side of the river, and pinned between two Targaryen-loyal armies, the Faith Militant forces were massacred, in the most lopsided battle Westeros had seen since the Field of Fire. So many men were killed that the Mander River ran red with blood for twenty leagues, and Stonebridge was known as "Bitterbridge" ever after. The Poor Fellows had been commanded by a huge man known as Wat the Hewer, who did manage to kill half a dozen knights of the royal army, even killing the royal host's commander Lord Meadows of Grassy Vale. Wat was captured alive and brought to King's Landing in chains.
The second prong of the Faith Militant's offensive further north fared somewhat better, though in the sense that it was a well-organized military campaign that at least managed to inflict serious damage against the Targaryens for a time. This second, larger host was composed of 13,000 Poor Fellows, and as they advanced east along the Goldroad they were joined by 200 well-armed and mounted knights of the Warrior's Sons from Stoney Sept. Along the way, they were joined by household knights and feudal levies of secular rebel lords, until the host numbered a full 20,000 men, with hundreds of well-armed knights. The army was led by Lord Rupert Falwell, famed as the Fighting Fool - due to the fact that House Falwell is a noble House from the Westerlands whose sigil is actually a court fool in red and gold motley against a black plain (the name was not meant derisively). They met in battle with the main Targaryen army at the fork of the Blackwater river, where the Goldroad crosses the tributary of the Blackwater coming south from Gods Eye lake, right before it meets the other major tributary coming east from Stoney Sept. The royalist army was only the same size as the rebel army, due to being assembled on short notice, but it had twice as much heavy horse and a large contingent of archers - as well as Maegor himself, riding Balerion. The rebels also picked the most opportune time to join battle, during a driving rainstorm: as seen with Meraxes during the Last Storm in the War of Conquest, it is more difficult for dragons to fly and to project their fire-breath during a rainstorm. This does not outright quench a dragon's fire, however, it simply dampens it. Balerion was still able to burn quite a few men. Still, the rebels picked as good a time as ever to engage a royal army led by a massive dragon. This battle did not turn out to be the lopsided boot-stomp that the royalists enjoyed at Bitterbridge, but was instead a savage slaughter. The Fighting Fool died in the battle, but not before he cut down two members of the Kingsguard. The Targaryens prevailed, but their army took extensive casualties.
This was not the end of the Faith Militant uprising, unfortunately, but only the beginning. The war shifted into a new phase, actually more of a series of ongoing but related uprisings, settling into a bloody pattern: the Targaryens would crush any rebel army that openly rode against them, but take heavy casualties in the process, while the Faith continued to raise new conscripts to the Poor Fellows from the peasantry. Most of these untrained mobs would be slaughtered, but it kept the fight going, and they were frequently bolstered by rebel lords committing their trained hosts of mounted knights to the conflict. The uprisings would continue throughout the rest of Aenys I's reign - though by this point, the real power was functionally controlled by Maegor as Hand of the King.
King Maegor the Cruel (r. 42-48 AL)
After Aenys I died in 42 AL, he was succeeded by his half-brother Maegor, whom subsequent generations remember as "Maegor the Cruel". It is uncertain how Maegor was able to achieve this given that Aenys already had three children, but it might not have been an outright usurpation. Maegor had tactfully married Aenys's oldest child, his daughter Rhaena. Aenys did have a son, but Jaehaerys was only a child, and the Targaryens were beset by massive uprisings against their rule. Given that the Targaryens had not officially settled upon using the inheritance system of the Andals of Westeros, enough were apparently willing to accept Maegor's succession: the other option was to crown the child Jaehaerys king in the midst of massive rebellions, Maegor was already the one effectively crushing each new rebel uprising as it appeared, and he was married to the previous king's oldest daughter anyway (his own half-niece) so the bloodlines were secure.
Maegor aggressively embraced the Targaryen practice of incestuous marriage. His first marriage had been to his half-sister, Aenys's full sister, but after that marriage proved childless he remarried to Aenys's daughter. He may have also married Aenys's other sister. What is known is that Maegor married a full six times, and had many of his wives killed when they could not produce any heir for him (leading one to suspect that the biological failure was probably in Maegor, not his wives). Some alternate sources say he had as many as nine wives. One of his wives was not a Targaryen but a woman named Jeyne from House Westerling in the Westerlands. As a grim irony, there wasn't really any need for Maegor to kill his wives: alone of all Targaryens after Aegon I died, Maegor tried to continue their practice of polygamous marriage. Thus he didn't actually need to kill his previous wives in order to remarry to new ones: he was married to many of them at the same time, and simply killed the previous ones later on out of petty anger over their perceived failure.
Maegor was a huge, well-muscled bull of a man, quick to aggression. He had three separate Grand Maesters executed during his reign. The Red Keep was finally finished during his reign: the massive royal castle which dominates the skyline of the capital city. Most of the rest of King's Landing had also finished construction and become functional by this time. Maegor had all of the designers and builders of the Red Keep killed after it was completed, to preserve the secrets of all of the hidden passages that honeycomb it. "Maegor's Holdfast" is named in his honor: the sturdy castle-within-a-castle which is the most secure section of the Red Keep, where the royal apartments are located.
Maegor's brutal crackdown against the Faith Militant became more ruthless than ever, as he made a royal proclamation of a bounty on their members: a Gold Dragon for the scalp of a Warrior's Son, and a Silver Stag coin for the scalp of a Poor Fellow. It was for this specific action he earned the sobriquet "Maegor the Cruel".
Some historians say that Maegor only exacerbated the Faith Militant uprisings with his brutal crackdowns where diplomacy might have sufficed, and his ruthless tactics only turned more men against him. Other historians point out that many of the uprisings, while latching onto the overall series of uprisings in defense of the Faith, were truly just local lords who bore a grudge against the new Targaryen kings, and ideology had nothing to do with their opposition to the Iron Throne. In the latter case, Maegor was perhaps a brutal man well-suited to a brutal time, when the Targaryens were still uneasy in their rule over Westeros and surrounded by powerful lords who not long ago ruled as kings, and needed to be brought to heel.
King Jaehaerys I (r. 48-103 AL)
Whatever the case, Maegor died after six years of rule, while seated on the Iron Throne. Some legends say he was killed by cutting himself on the throne (whichis unlikely), while others suspect that there is a good chance he was poisoned. Despite all of his efforts Maegor died childless, and the crown passed to Aenys I's son Jaehaerys (given that his older sister Rhaena apparently died by this point, possibly killed by Maegor).
King Jaehaerys I was an intelligent and wise ruler, and became known as "Jaehaerys the Conciliator" when he negotiated a peaceful end to the Faith Militant uprising. The rebellions had been going on for roughly eight years by the time Maegor died, and while he had crushed their armies at every turn he had not succeeded in breaking their will to resist another day. Jaehaerys resolved the issue by offering generous terms of a blanket amnesty, if in return the Faith of the Seven would disband its military orders. He also came to terms with the Faith over Targaryen incestuous marriages: the Faith didn't have to praise Targaryen incest practices, it simply had to grudgingly tolerate it. These were probably the best terms the Faith could have hoped for, so it gladly accepted, and the uprisings finally ended.
By this point all remaining Targaryens were the product of Aegon and his sister-wives so all would be accused of having incestuous bloodlines no matter what they did. Jaehaerys I continued the practice by marrying his own sister, Alysanne.
Jaehaerys I is fondly remembered as one of the best kings the Targaryen dynasty ever produced, and his long reign became synonymous with peace and prosperity. He was the longest-ruling king in history, sitting on the Iron Throne for fifty-five years, and due to his advanced age by the end of his reign is often called "the Old King". As his Hand of the King, Jaehaerys chose Septon Barth, a smith's son who served as Hand for forty years. Barth was just as skilled of an administrator as Jaehaerys, also fondly remembered as one of the best Hands ever to hold the office. Together, they brought the Seven Kingdoms into an unprecedented era of peace and prosperity which lasted for decades. So skilled was Jaehaerys I as a negotiator that he even managed to negotiate a truce in the centuries-old feud between House Bracken and House Blackwood, which would last for several generations.
Meanwhile, Alysanne was also actively involved in court affairs and became one of Jaehaerys's chief counselors, if not practical co-ruler. Alysanne also rode her own dragon, named Silverwing. She is also fondly remembered in the Seven Kingdoms, as "Good Queen Alysanne".
At least six dragons were alive during Jaehaerys I's reign, when he and Alysanne rode them on a formal visit to Winterfell. Among them was Silverwing, Alysanne's dragon, which she rode on a whim further north to visit the Wall. Jaehaerys himself followed shortly afterwards. Alysanne was so impressed with the bravery of the Night's Watch that she successfully convinced Jaehaerys to double the lands of the Gift, which had supported the Watch for thousands of years, by extending it further south.
The new era of peace that began when Jaehaerys I was crowned in 48 AL ultimately lasted for eighty years, throughout his long and wise reign, and after his death in 103 AL, under his successor Viserys I. In many ways this was the Golden Age of the Targaryen dynasty: entire generations were born and died under the peace and prosperity of Targaryen rule. The petty wars between the Seven Kingdoms in past centuries were gone, and the rebellions surrounding the establishment of the Targaryen kingship had finally ended.
Not much is said about this long era of peace, as history books tend to be filled with descriptions of bloody wars, allegedly exciting conflicts, and mass suffering. In time this era of peace did come to an end, in the destructive Targaryen civil war following Viserys I's death known as the Dance of the Dragons.