Following the defeat of Prince Garin the Great by the dragonlords of Valyria, Princess Nymeria led her people into every ship, skiff and raft they had and fled Westeros. The Rhoynar sailed across the Narrow Sea until they eventually reached Dorne, burning all ten thousand of the ships afterwards.
In the books
In the A Song of Ice and Fire novels, the Rhoynish migration occurred about a thousand years before the events depicted in the novels. Princess Nymeria and the surviving Rhoynar traveled from Essos to Dorne on ten thousand ships following their defeat by the Valyrian Freehold in the Second Spice War.
Prince Garin the Great of Chroyane led the Rhoynar to several victories against the Valyrians, only to be defeated by 300 dragons commanded by the dragonlords of Valyria. Two hundred and fifty thousand men were killed and Nymeria evacuated the surviving Rhoynar, most of whom were women, children and the elderly, across the Narrow Sea, allegedly on ten thousand ships.
Following unsuccessful attempts to settle in the Basilisk Isles, Sothoryos, Naath and the Summer Islands, they Rhoynar landed at the mouth of the Greenblood in Dorne, though some ships landed on the Stepstones or surrendered to slavers from Lys and Tyrosh. Nymeria found common cause with Mors Martell of Sunspear and married him, burning the ships to affirm that she could could not return to Essos. By joining their forces, Nymeria and Mors were able to unite the entire peninsula of Dorne in Nymeria's War.
Nymeria and her ten thousand ships are depicted on a tapestry in a secret room in Sunspear's shadow city.