Seven Kingdoms of Westeros are mentioned a lot of time in the game of thrones tv series. If you count them of the map, you will find there are actually 9 kingdoms. Then why are they calling it seven kingdoms?
When Aegon I Targaryen conquered the Westeros with 3 dragons Balerion, Vhagar and Meraxes, there were actually 7 independent realms formed by the First Men.
Aegon I Targaryen unified 6 kingdoms of Seven Kingdoms of Westeros except Dorne. He could not conquer Dorne which resist Targaryens for couple of centuries.
Seven Kingdoms of Westeros are as follows
Kingdom of the North
The kingdom in the North is ruled from the Winterfell by the King or Queen in the North. Current ruler is the House Stark. When Aegon I Targaryen conquered the Westeros, the kingdom of North made to serve the Iron throne.
It was restabilised by Robb Stark after the execution of his father Eddard Stark. But he was defeated by the Iron throne.
The kingdom once again restabilised by Jon Snow (game of thrones) as the king in the North.
Jon Snow (game of thrones) pledged his loyalty to the Daenerys Targaryen. So, it again became the part of Seven Kingdoms of Westeros.
It once again restabilised by Sansa Stark when Brandon Stark became the new King of the Andals.
Kingdom of the Mountain and the Vale
Before the War of Conquest (Aegon I Targaryen conquered the Westeros), it was independent part of Seven Kingdoms of Westeros.
The descents of the Mountain Kings ruled the Vale from the Eyrie.
Ser Artys Arryn became the new King after Andal Invasion when last ruler of the Vale lost to him and he founded House Arryn.
At the time of Aegon I Targaryen conquest, Ronnel Arryn King of the Vale (Kingdom of the Mountain and the Vale) was young, so her mother ruled it. Visenya Targaryen flew her dragon over the castle of Vale, Ronnel asked her to give him ride on dragon which she did. Seeing king on her lap on dragon, her mother surrendered and Arryns continued to rule under the Targaryens. Now it was not independent part of Seven Kingdoms of Westeros.
Kingdom of the Isles and the Rivers
The kingdom was independent part of 7 kingdoms of Westeros under Harren Hoare with capital Harrenhal.
House Hoare conquered all of Riverlands and founded Kingdom of the Isles and the Rivers three generations before the Targaryens destroyed its capital Harrenhal with Dragons and removed its existence.
Kingdom of the Rock
House Casterly founded the Kingdom of the Rock at the time of First Men. It was independent part of 7 kingdoms of Westeros until Aegon conquered it.
House Lannister started ruling it after the Andal Invasion. Loren Lannister fought against the Targaryen Conqueror with the help of king of reach but surrendered to Aegon and Kingdom of the Rock became part of the unified seven kingdoms of Westeros.
Kingdom of the Stormlands
Stormlands Kingdom was founded by the first Storm King Durran Godsgrief. It was independent part of the 7 kingdoms of Westeros.
It is said that the kind had taken Elenei, the wind goddess and built castle of Storm’s End.
The kingdom fell when last king was killed during the War of Conquest.
Kingdom of the Reach
Garth Greenhand founded the kingdom. It was independent part of the kingdoms of the Westeros.
The kingdom fell under Iron throne after attack of Aegon. House Tyrell became new ruler of the kingdom of reach.
Principality of Dorne
The Principality of Dorne was independent part of Westeros and during the conquest of Targaryens, it remained unconquered. Targaryens attacked again and took it for four years.
Dorne fought against it in restored it once again. Dorne united with the Iron throne by marriage-alliance on equal terms.