Andal Invasion

The Andal Invasion was the migration of the Andals to the continent of Westeros from their homeland of Andalos on the eastern continent of Essos, six thousand years before the War of the Five Kings. The migratory invasions of the Andals occurred in waves over many centuries, but by the end they had killed or conquered all of the First Men south of the Neck. By the end of the invasions, the Andals had displaced the First Men as the major ethnic and cultural group of Westeros, to the point that peoples living in Essos frequently refer to Westeros as the Land of the Andals.

Prelude
The Andals believe that the God of Seven appeared to them, and told them to conquer the promised land in Westeros.

Invasion
They started in the Vale but gradually conquered almost everything.

Aftermath
The North held out, but otherwise the Andals became the dominant cultural group in Westeros.

The Iron Islands are somewhat of an odd case, in that the Andals who conquered it "went native" and just assimilated into the distinctive Ironborn culture of seafaring and piracy. They even abandoned worship of the Seven and adopted the local religion of the Drowned God. Thus the Ironborn are ethnically similar to the Andal/First Men mix that the rest of the South possesses, but they are culturally distinct.

Dorne was also conquered by the Andals and they intermingled with the First Men there, but they remained as numerous petty fiefdoms and didn't unify.

In the books
The Andal Invasion is sort of like the Anglo-Saxon heptarchy. Unlike England, they weren't politically unified when the Targaryens (their Normans) invaded. Dorne was only later unified when the Rhoynar migrated there and intermingled with the Andals and First Men living there. The Andals there live in the mountains not the coasts.