a song of ice and fire - What are the different provinces mentioned in a Game of Thrones?

At one point in Game of Thrones, it says that no longer are there Seven Kingdoms, since they have been united, but now there are provinces:

That was close on three hundred years ago, when the Seven Kingdoms were kingdoms, not mere provinces of a greater realm.

What are the names of these provinces, and which houses control them?

Answer

It really depends on how you define provinces. But it is mostly agreed that Westeros is divided into these large chunks. Note: this is the political situation at the beginning of Game of Thrones only.

1. The North. Which pretty much includes everything north of the Neck and south of the Wall. It is ruled by House Stark from their seat at Winterfell.


2. The Riverlands. South of the Neck and encompassing the fertile area containing much of Westeros's rivers (hence the name). It is ruled by House Tully from their seat at Riverrun.


3. The Westerlands. The area to the west of Westeros. It is ruled by House Lannister from their seat at Casterly Rock.



4. The Iron Islands. Seven large islands to the north west of Westeros.. They are ruled by House Grayjoy from their seat at Pyke.


5. The Vale of Arryn. A mountainous area to the east of Westeros. It is ruled by House Arryn from their seat at The Eyrie.


6. The Crownlands. The area surrounding the capital, King's Landing. It is directly ruled by the Crown (at the time King Robert Baratheon, first of his name ... etc etc.)


7. The Reach. Fertile area to the south west of Westeros. It is ruled by House Tyrell from their seat at Highgarden.


8. Dorne. Arrid area to the south east of Westeros. It is ruled by House Martell from their seat at Sunspear.


9. The Stormlands. Coastal area between Dorne and the Crownlands. It is ruled by House Baratheon from their seat at Storm's End.

Each of these lands and their respective Great Houses rule over a large number of feudal lords who have their own lands within the greater province.