The Seven Kingdoms confusion
Seven Kingdoms were the political entities which existed at time of Aegon's landing on shores of Blackwater Bay. Number of these political entities has always varied from time to time i.e. During the Age of Heroes and Andal invasions, there were hundreds of small Kingdoms in Westeros. Before Aegon's landing, there were only seven however.
The titles which Aegon created for himself and all other Kings inherited after him were:
King of the Andals and the Rhoynar and the First Men, Lord of the
Seven Kingdoms, Protector of the Realm
This title remained the same even 300 years after Aegon's death despite the fact that in Post-invasion era, there were no longer 7 Kingdoms or administrative Units. This is the title which causes all the confusion because many people do not realize that When Aegon came, there actually were seven Kingdoms which is no longer the case today or necessarily before Aegon's times.
De-Jure Kingdoms
The De-jure Kingdoms of Westeros which are based on distinct cultures and religions (And in some cases ethnicities) are:
- Kingdom of Stormlands
- Kingdom of Riverlands
- Kingdom of Vale and Mountain
- Kingdom of Iron Islands
- Kingdom of the North
- Kingdom of the Reach
- Kingdom of Westerlands
- Principality of Dorne
That gives us a sum of 8 De-jure Kingdoms.
Historical Aspect and De-Facto Kingdoms
When Aegon came however, Westeros was not divided along de-jure lines. Kingdoms were continuously at wars against each other with all of them occupying de-jure regions of some other Kingdom.
Many centuries before the conquest, House Justman ruled Riverlands with splendor. Last of their line Bernarr II Justman was killed by King Qhored Hoare of Iron Islands. Riverlands fell into chaos with every House claiming the throne for themselves. The Chaos Ended when King Torrence Teague claimed the throne. Riverlords did not like House Teague and rose in rebellions many time.
Finally during one uprising, The traitorous Tullys, Vances and Blackwoods under leadership Roderick Blackwood invited King Arlan III Durrandon of Stormlands to attack their own liege King Humfrey Teague. House Teague was wiped out in Battle of Six Kings and King Arlan merged Riverlands into Stormlands.
Years passed until Harwyn 'Hardhand' Hoare took throne of Iron Islands. The ambitious King kicked Stormkings out of Riverlands and became King of Isles and Rivers. Rule of House Hoare over Riverlands continued until Aegon came to conquer Westeros.
So at time of Aegon's conquest, Riverlands was not a royal political entity but rather subjects to Kingdom of Iron Islands. De-Facto Kingdoms at that time were the following (With their current status in parenthesis):
- Kingdom of the North (Still Exists)
- Kingdom of the Reach (Still Exists)
- Kingdom of Stormlands (Still Exists)
- Kingdom of Isles and Rivers (Non-Existant: Split into two)
- Kingdom of Westerlands (Still Exists)
- Kingdom of Vale and Mountain (Still Exists)
- Principality of Dorne (Still Exists)
A Pre-conquest map of Westeros from Crusader Kings II: A Game of Thrones Mod (Which I can verify is correct except the fact that technically Dorne should be referred to as a principality as Rhoynar culture has Princes, not Kings even though they are equivalents of Kings for all intents and purposes, and that It should be Kingdom of Isles and Rivers instead of Kingdom of Iron Islands):

Administrative Units of Post-Conquest Era
Aegon the Conqueror usually retained the political hegemony of Kingdoms which submitted to his rule.
The Notable exception was however Kingdom of Isles and Rivers because they refused to submit and chose to fight till the bitter end. Riverlords under Lord Tully rallied to Aegon's cause against King Harren the Black. Harren the Black refused to submit to Aegon's rule and chose to fight. After House Hoare was extinguished in Burning of Harrenhal, Aegon raised Lord Edmyn Tully to office of Lord Paramount of Riverlands due to Edmyn Tully's service in cause of Targaryens especially in the initial days of the conquest when Targaryen position was uncertain.
The other Part of Harren's Kingdom, The Iron Islands fell into chaos. Many pretenders to House Hoare's inheritance rose and fought against each other. King Aegon invaded Iron Islands two years after completion of his invasion of mainland Westeros (With Exception of Dorne). Aegon defeated all the pretenders but allowed them to choose their own Liege Lord which was unique as all other Lords Paramount were appointed by Aegon I and Ironborn were the only people who were allowed to choose their own Leaders. Defeated Ironborn chose Lord Vickon Greyjoy. Aegon named Vickon as Lord of Iron Islands and left for King's Landing. Greyjoys were now Lord Paramounts of Iron Islands and answerable to no one except the King himself.
Other than that Aegon also took the disputed regions of Blackwater Rush and Bay as royal lands because there was no clear owner of those Lands and Aegon needed a foothold on main continent to stabilize his rule. This region was named Crownlands.
So in Aegon's time there were following administrative units with their own governing ruler:
- Paramount Lordship of Vale and Mountain
- Paramount Lordship of Iron Islands
- Paramount Lordship of Westerlands
- Paramount Lordship of North
- Paramount Lordship of the Reach
- Paramount Lordship of Stormlands
- Paramount Lordship of Riverlands
- Royal fief of Crownlands
Later Dorne also came under Iron Throne during reign of King Daeron II (And briefly in reign of King Daeron I). According to the agreement, Dornish retained several special rights which included the right to address their rulers as Prince/Princess instead of Lords. So the new Administrative units were:
- Paramount Lordship of Vale and Mountain (Currently ruled by House Arryn)
- Paramount Lordship of Iron Islands (Currently ruled by House Greyjoy)
- Paramount Lordship of Westerlands (Currently ruled by House Lannister)
- Paramount Lordship of North (Currently ruled by House Stark in Show, House Bolton in Books)
- Paramount Lordship of the Reach (Currently ruled by Olenna Tyrell nee Redwyne in Show. House Tyrell in Books)
- Paramount Lordship of Stormlands (No current Lord Paramount in show, "House Baratheon 😉" in Books)
- Paramount Lordship of Riverlands (Currently ruled by House Frey in show, House Baelish in Books)
- Royal fief of Crownlands (Currently ruled by House Lannister in show, House Baratheon (wink wink) in Books).
- Principality of Dorne (Currently ruled by a bastard daughter of Lord Uller in show, House Martell in Books)
A map of post-Robert's Rebellion Westeros which has the same administrative units as constituted by Aegon the Conqueror.
