The Kingdom of Mountain and Vale was one of the independent kingdoms established in the continent of Westeros prior to the War of Conquest. During the time the First Men controlled the continent, the dynasty of the Mountain Kings ruled the Vale from their stronghold, the Eyrie. However, the last First Men ruler of the Vale, the Griffin King, lost the Vale to Ser Artys Arryn, one of the leading figures of the Andal Invasion. Ser Artys was declared King of the Mountain and the Vale and founded House Arryn, a line of Andal nobility that ruled the Vale until the invasion of Aegon the Conqueror.
At the time of the Targaryen Conquest, the last King of the Mountain and the Vale was a young boy, Ronnel Arryn, so the Vale was ruled on his behalf by his mother, Queen Regent Sharra Arryn. Ultimately Visenya Targaryen flew over the guarded mountain passes on her dragon, Vhagar, directly to the Eyrie. Visenya did not burn the castle, and when the young King Ronnel came out to meet her in the courtyard he asked if she would give him a ride on her dragon, and she agreed. When Sharra arrived and saw her son in Visenya's lap on the dragon, the implicit threat was enough to make her surrender the castle.
In return for their submission, the the Arryns were continued to rule the region as Lords Paramount of the Vale - under the Targaryens.
In the books
In the A Song of Ice and Fire novels, the history of the Kingdom of Mountain and Vale is the same. However, during the rule of House Arryn, which had ended the line of the Mountain Kings, the kingdom was renamed the Kingdom of Vale and Sky.