Jaehaerys the Conciliator and the long peace
After Maegor the Cruel's short and appalling reign, Jaehaerys I and his queen Alysanne gave the realm fifty-five years of peace ÔÇö roads, laws, an end to the Faith's war on the crown, and more surviving children than any Targaryen before or since. The dynasty's high-water mark, and the reason the Great Council of 101 AC could settle a succession by debate rather than by dragon.
The fuller tale
Between Aegon and Jaehaerys sits Maegor, who is remembered as the Cruel by a realm that does not hand out that name lightly: a six-year reign of burnings, a war against the Faith, three of his own nephews dead, and an end on the Iron Throne itself with the barbs in him and nobody able to say for certain whether he had been murdered or the chair had simply had enough.
Jaehaerys came to the throne at fourteen and reigned fifty-five years, which is longer than any other Targaryen and longer than most of them managed to stay sane. His first act was to make peace with the Faith: he gave up nothing that mattered, disbanded the militant orders, and in exchange the Seven stopped preaching against his family's marriages. It cost him a promise and saved his successors a civil war.
With his queen Alysanne he did the unglamorous things that no song remembers. The roads ÔÇö the kingsroad, the roseroad, the ocean road ÔÇö were driven under him, and they are why a raven and a rider can cross the realm in this story at all. The laws were gathered and written down in one code by his Hand, Septon Barth, and it is still the law three hundred years later. The right of the first night was abolished, over the loud objection of lords who enjoyed it.
Good Queen Alysanne
Alysanne flew Silverwing and did not stay at court. She went north, sat with the Night's Watch, was appalled by how little land they had to feed themselves on, and talked her husband into doubling the Gift. She held women's courts wherever she went and listened to complaints no king would have heard. When Jaehaerys named their second son heir over their elder daughter's line, she left him over it for two years, which is the single most quoted domestic quarrel in Westerosi history and a preview of the war that arrives two reigns later.
The Great Council of 101
Jaehaerys outlived his heirs, and rather than pick a successor himself he called every lord in the realm to Harrenhal to decide it. Fourteen claimants were heard; the lords chose Viserys, grandson through a son, over Rhaenys, granddaughter through an elder son. A precedent was made that afternoon by vote rather than by dragon: in Westeros, a male line jumps a female one.
It was the high-water mark of the dynasty and the seed of its worst war, and it is a fair summary of the whole Targaryen century that both of those are true at once.
Elsewhere in the wiki
 See the world as it stood in 48 AC What became of it  if an ending is written, it is here. Spoilers.
The chronicle records no ending here ÔÇö or none the maesters have set down.