The Targaryens, blessed with the best hair, the best pets, and the best lineage, were wronged in the Game of Thrones series, but the spinoff show House of the Dragon did them justice. Fans saw the ruins of the dynasty in the former, but the latter made sure everyone saw what Daenerys was fighting so hard to restore.
But as it is said that not everything that glitters is gold, similarly, not every thriving dynasty is as stable as it seems. Where there is power, there are relatives who are plotting to kill each other to gain that power. And when it is the House of the Dragon and the Targaryens are involved, then the result would be fire and blood everywhere.
As seen in the House of the Dragon, a war of succession broke out between siblings, Rhaenyra Targaryen and Aegon Targaryen (not the original one), and the result was known as the Dance of the Dragons. Naming a war starting with the word 'dance' didn't make it cool, as the war was brutal.
So, let's dive right in to know what all went down.
What was the Dance of the Dragons in House of the Dragon?

Before you learn about the actual event, you need to know who the key players were involved from the House of the Dragon.
King Viserys I: The head of the House of the Dragon and main culprit who started the mess by creating power-hungry children was King Viserys I. He was a sweet and just king and was known as Viserys the Peaceful. All he ever wanted was for everyone to get along and live peacefully. Unfortunately for him, his kids were his opposites. All they wanted was chaos.
Princess Rhaeneyra: Rhaeneyra was Viserys' oldest child from his first marriage and was named the heir to the Targaryen throne. She was bold, tough, and an amazing dragon rider, but her family dynamics changed when her best friend married her father.
Alicent Hightower: Alicent Hightower, later known as Queen Alicent, was Rhaenyra's BFF, Viserys' second wife, and mother of Prince Aegon II, Prince Aemond, and Princess Helena Targaryen. She was big on tradition and felt that her older son, Aegon, was the rightful heir to the throne instead of Rhaenyra.
Prince Aegon II: Aegon is Alicent's first-born and Viserys' second-born child. He is a walking red flag and walks like his dad owns the place (which he kinda does). He knows he is royalty and is not afraid to flaunt it. He wasn't particularly interested in ruling until his mother told him to do so.
Viserys wanted to be progressive and name his oldest child, a daughter, his heir, but it didn't sit well with the rest of Westeros.
King Viserys I Targaryen's Will

In House of the Dragon, King Viserys I didn't just name Rhaenyra his heir; he also made everyone pledge their loyalty to her. They were made to bend their knees to her and sing songs in her honor. But one person thought it wasn't okay, and it was Queen Alicent. She had given birth to not one but two male Targaryens and wasn't going to let them be wasted.
She opposed the decision and convinced many of her husband's associates that her son, Prince Aegon, was the true heir because he was a man and had a sword. Many of the associates saw the logic in this and decided to switch sides. Up until King Viserys I was alive, no one questioned his decision to name Rhaenyra his heir, but the minute he was gone, everything changed.
Queen Alicent's Misunderstanding

In House of the Dragon, as King Viserys I's health slowly started to deteriorate, his speech was losing volume, and his mental health was failing. On his deathbed, while being high on the milk of the poppy, he mistook his wife, Alicent, for his daughter Rhaenyra and muttered something that she completely misunderstood.
He was talking about Aegon the Conqueror's prophecy, where he said that the Targaryens needed to unite against the White Walkers. Alicent, who didn't hear anything beyond the word Aegon believed that he was talking about her son Aegon II and that he should be king.
That was enough for Alicent to crown her son as king and gaslight anyone who said otherwise. She had her horde of supporters led by her dad, the Hand of the King, and together they placed Prince Aegon II on the throne.
Targaryen vs. Targaryen

As soon as King Viserys I died, Queen Alicent lost no time in usurping the throne for her son. She cut short the mourning of the King and made everyone prepare for Prince Aegon II's coronation. They locked down the Red Keep and kept the news of her father's demise a secret from Rhaenyra, who was chilling at Dragonstone. This resulted in the division of the House of the Dragon.
Anyone who opposed their operation was either imprisoned or turned into dinner for the dragons. Alicent and her friends, the Greens, were working faster than the speed of light to make Aegon II's coronation a success. The same Aegon who would rather be at a drunken debauch than a king.
Rhaenyra was heavily pregnant and had no idea that her inheritance was being gifted to her unworthy younger brother. The Greens used Aegon the Conqueror's crown for the coronation and paraded Aegon II before the public, and thus, Alicent's mission was accomplished.
Blacks vs. Greens

When the news finally reached Rhaenyra, it hit her hard. She received three heartbreaking news at once. Her father was dead, her throne was stolen, and people who pledged their loyalty to her had switched sides. Amid all this, she had a miscarriage.
Since Rhaenyra was a Targaryen, she wasn't one to let her grief turn into depression; instead, she turned it into anger. Her first order of business was to name herself queen with her father's actual crown, which was smuggled to her by Ser Steffon Darklyn.
Then she gathered her loyalists, called the Blacks, found herself some healthy dragons, and waged a war against those who betrayed her. Thus, the Dance of the Dragons began.
Now, the House of the Dragon has two monarchs with enough dragons to turn Westeros into a crispy chicken platter, and enough energy for wrong decisions to ruin it for future generations.
So, the answer to the question of what started the Dance of the Dragons in House of the Dragon? The answer is boiling envy inside family members and a deathbed murmur gone wrong. Alicent's confusion upon hearing the word 'Aegon' started the whole war and resulted in a lot of destruction.
The war resulted in the deaths of allies, family members, and dragons. It almost resulted in the extinction of House Targaryen, and all this because no one corrected Alicent at the right time.