The Dawn Age ÔÇö the children of the forest
Before men, Westeros belonged to the children of the forest: a small, dark-eyed people who carved faces into the weirwoods and sang a song of earth. They kept no cities and wrote no books, and everything we think we know of them comes to us through thousands of years of retelling ÔÇö which is to say, through the mouths of the people who later took their land.
The fuller tale
There is no reliable account of the children of the forest, and there cannot be one. They kept no annals, carved no inscriptions anyone has learned to read, and built nothing that survives above ground. Every word written about them was written thousands of years afterward, in a language they did not speak, by the descendants of the people who took their country. A maester reading his own order's histories on this subject is reading a rumour that has been copied out four hundred times.
What can be said with some confidence is negative: before men, Westeros was forest, and the forest had inhabitants who were not men. The weirwoods with faces are real and are older than any castle. The obsidian arrowheads turned up by northern ploughs are real. The caves under the hills that nobody dug are real. Everything past that ÔÇö the size of them, the length of their lives, the singing, the greenseers ÔÇö comes down through the mouths of their enemies and should be held loosely.
The account the maesters settled on describes a small, dark, long-lived people, brown-skinned and large-eyed, dressed in leaves and bark, armed with weapons of dragonglass because they had no metal. They lived in the deep wood and under it. They had no kings that anyone could identify and no cities to burn, which is one reason the war against them took so long: there was never a capital to take or a crown to knock off.
Their gods had no names and no faces except the ones carved into the trees, and they became the gods of the men who killed them ÔÇö which is the strangest thing in this whole strange chapter. A northern lord kneeling in his godswood today is praying at an altar built by a people his ancestors hunted to the edge of the world.
The song of earth
The one power consistently attributed to them is the one hardest to test: greenseeing, the ability to look through the eyes of a weirwood, and through the eyes of beasts, and ÔÇö in the most extravagant tellings ÔÇö through time. The Citadel's position for three centuries was that this is a story told about a story. That position has not aged well for anyone who has followed the tale to its northern end.
The other power the songs give them is the Hammer of the Waters: the breaking of a landmass by will. It is invoked twice, at the Arm of Dorne and at the Neck, and both times it is used to stop an invasion by drowning the road it came in on. Whether that is magic or a memory of an ordinary flood dressed up over ten thousand years is a question the sources cannot settle. What the map shows is that the Arm is broken and the Neck is a swamp, and both of them happen to sit exactly where an army would have to cross.
Where they went
They did not vanish in one generation. They were pushed, and then pushed further, and the pushing took so long that each generation of men could reasonably believe the children had always been a story. By the coming of the Andals they were gone from the south; by the Conquest they were a thing the North still half-believed in and the south laughed at; by the end of this chronicle there are perhaps a few dozen left, in a cave beyond the Wall, keeping a very old promise.
Elsewhere in the wiki
 See the world as it stood in 12,000 BC 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.