King of the Andals, the Rhoynar, and the First Men

The King of the Andals and the First Men is the ruler of the Seven Kingdoms, the unified realm which includes almost all of Westeros, except the lands Beyond the Wall in the frozen north. The King's complete titles are King of the Andals and the First Men, referring to the two largest ethnic groups in the continent, Lord of the Seven Kingdoms, referring to the seven independent kingdoms that existed prior to their unification in the War of Conquest, and Protector of the Realm, referring to the King's duty of maintaining peace, order and justice throughout the realm.

The office of the ruler of the Seven Kingdoms is often referred to as the Iron Throne, in reference to the eponymous throne in which the King holds court.

The position was created when Aegon the Conqueror succeeded in his conquest of Westeros, unifying the independent kingdoms of the Isles and the Riverlands, the Rock, the Reach, the Mountain and Vale, and the Kingdom of the North. The Princedom of Dorne was later united to the realm through marriage alliance.

The King is formally addressed by his subjects as "Your Grace" and in official events referred to employing the following structure: "Name" of the House "Name" the "ordinal number" of His Name, King of the Andals and the First Men, Lord of the Seven Kingdoms and Protector of the Realm". For example, Robert Baratheon is formally referred to as "Robert of the House Baratheon, the first of His name" etc.

Targaryen Dynasty

 * {Aegon I Targaryen}, called "Aegon the Conqueror" and "Aegon the Dragon", the founder of the Targaryen Dynasty and the Iron Throne.
 * {Maegor I Targaryen}, called "Maegor the Cruel"
 * {Aegon II Targaryen}
 * {Rhaenyra Targaryen} - rival claimant to the Iron Throne against her half-brother Aegon II. Retroactively considered to not be the lawful heir, though because Aegon II died without a male heir, Rhaenyra's own son ended up succeeding him as Aegon III.
 * {Aegon III Targaryen}, called "the Dragonbane". Son of Rhaenyra.
 * {Baelor I Targaryen}, called "Baelor the Blessed" and "Baelor the Beloved".
 * {Viserys II Targaryen}
 * {Aegon IV Targaryen}, called "Aegon the Unworthy"
 * {Daeron II Targaryen}, called "Daeron the Good"
 * {Aerys I Targaryen}
 * {Maekar I Targaryen}
 * {Aegon V Targaryen}, called "Aegon the Unlikely", as he was the fourth son of a fourth son yet still inherited the throne. Younger brother of Aemon Targaryen, who voluntarily refused the crown as he had already joined the order of Maesters.
 * {Aerys II Targaryen}, called "the Mad King" and "King Scab". Killed by Ser Jaime Lannister during the Sack of King's Landing.

Baratheon Dynasty

 * {Robert I Baratheon}. Became king after Aerys II Tararyen was overthrown and killed during Robert's Rebellion. Perished due to wounds sustained during a hunting accident.
 * Joffrey I Baratheon. Robert's heir. Legimitacy put into question.

Claimants
Four claimants to the Iron Throne emerged during and after Robert I's reign. Two of them claim the throne as their Targaryen birthright, while the other two claimed the throne after Joffrey's legitimacy was cast into doubt.
 * {Viserys III Targaryen} (son of Aerys II), called "the Beggar King" due to his impoverished exile in the Free Cities. He was killed by Khal Drogo and his claim passed to his sister.
 * Khaleesi Daenerys I Targaryen (daughter of Aerys II), called "Daenerys Stormborn", "Mother of Dragons", and "the Queen Across the Sea".
 * Stannis I Baratheon (younger brother of Robert Baratheon), called "the King in the Narrow Sea" due to his rule of Dragonstone.
 * {Renly I Baratheon} (youngest brother of Robert Baratheon), called "the King in Highgarden" due to his alliance with House Tyrell. Killed by a shadow.

Although Robb Stark of The North and Balon Greyjoy of the Iron Islands also declared themselves in defiance of Joffrey I, they are fighting to have their respective territories secede from the realm and form their own independent kingdoms, hence they do not claim the Iron Throne itself.

In the books
In the A Song of Ice and Fire novels, the full title used is "King of the Andals, the Rhoynar, and the First Men, Lord of the Seven Kingdoms, and Protector of the Realm". It's probable that "the Rhoynar" was omitted in Season 1 for the sake of not confusing viewers who haven't read the books with too much information, because Dorne and the Rhoynar haven't been introduced yet.

In the three hundred years between the Targaryen Conquest and the War of the Five Kings, there has never been a Ruling Queen: a female heir of the current monarch inheriting power in her own right. The first four Targaryen kings all had male heirs who were also their eldest child. However the fifth Targaryen king, Viserys I, only had one surviving child by his first wife before she died, a daughter named Rhaenyra. With no other heirs, Viserys I and his court raised Rhaenyra with the expectation that she would be the first Ruling Queen. However, Viserys I remarried late in life, and had several sons with his second wife, the eldest of which was his son Aegon II. When Viserys I died this sparked a succession war between Rhaenyra and Aegon II, known as the Dance of Dragons, which raged from 129 to 131 AL (about 170 years before the War of the Five Kings). Aegon II ultimately had Rhaenyra fed alive to his dragon, but her supporters continued to fight in the name of her children, and not long afterwards Aegon II himself died childless. As the only remaining heir of Viserys I or Aegon II, Rhaenyra's own son Aegon III inherited the throne (Aegon III's own sons both died childless, and ultimately Rhaenyra's younger son Viserys II succeeded to the throne). After the Dance of Dragons, the Targaryens revised the official royal succession laws to follow an extreme form of agnatic primogeniture, placing female heirs behind all possible male ones, i.e. if all of a king's sons died childless, his own younger brothers would inherit instead of his daughters (their nieces). Such was the case when after both of Aegon III's sons died childless, his daughter Daena was skipped over in succession for Aegon III's younger brother Viserys II. These altered inheritance laws ensured that there was no Ruling Queen in the history of the Seven Kingdoms. Many historians point to the succession of Rhaenyra's son Aegon III after Aegon II died as proof of the legitimacy of Rhaenyra's claim to inheritance in the civil war, and while she lived she did personally use the title of Ruling Queen. Officially, however, Rhaenyra is considered a rival claimant and is not counted in the formal line of succession. Any possible future Ruling Queen by the name of "Rhaenyra" would be titled "Rhaenyra I", not "Rhaenyra II". As this would lead to controversy over whether to acknowledge Rhaenyra's claim during the Dance of Dragons, subsequent generations of the Targaryen family simply avoided the issue by never naming any subsequent daughters "Rhaenyra".