Three Great Powers

The Three Great Powers (三大勢力), composed of the Marine Headquarters, the Shichibukai, and the Yonko, are the three greatest military forces on the oceans, and their relations are very vital to the World Government. The three powers exist in a delicate balance of power, not all the details of which are known. It is however known that only the other two organizations are equivalent to the Yonko.

Overview
Considering that these organizations are stated to be a balance and the fact that the loss of Crocodile threatened the balance, it is likely they are approximately equivalent in strength. The Yonko and Marines seem to be on the opposite ends of the spectrum, with the Marines being more orderly compared to the Yonko which are very chaotic, with the Shichibukai acting as somewhere in between.

The Gorosei fear that if this balance gets disrupted, the world itself could descend into anarchy, and the balance itself appears to be very fragile as the loss of even one key member (such as a Shichibukai) can disrupt the balance. This is why the defeat of Crocodile was so distressing to them; the loss of such a powerful ally in such a public manner tips the balance away from the Shichibukai and in turn the Marines. This leaves them with diminished force to counter the menace of the Yonko and allows the general populace to question their power and authority. Another concern is that the Marines are the sole organization completely under the supremacy of the World Government. The Shichibukai often follow orders on a whim, if at all, and the Yonko are entirely outside of the World Government's control.

As the delicate balance of the Three Great Powers is what stabilizes the world, the existence and actions of the Revolutionary Army and the Worst Generation (Luffy and the Straw Hat Pirates in particular) is a threat to that balance.

Summit War Saga
The arrest of Portgas D. Ace, the second division commander of the Whitebeard Pirates resulted in the Marines entering into hostilities with the Whitebeard Pirates. This eventually led to a major battle between the Marines and Shichibukai against the Whitebeard Pirates and their allies. This was the first known instance where any of the powers went into full-out conflict with one another. Prior to the battle, Kaido tried to attack Whitebeard and was intercepted by Shanks, causing a dispute between the Yonko. After Whitebeard's death, the balance collapsed, and the world entered into chaos, contrary to the belief that the great pirate's death would ensure peace.

At the conclusion of the Battle of Marineford, the Yonko have lost one from their ranks (Whitebeard), and the Shichibukai have lost four; Jinbe and Blackbeard resigned their positions (for different reasons: Jinbe because he opposed the war against Whitebeard and changed alliances to protect Fishman Island, and Blackbeard only joined for easier access to Impel Down - his ultimate goal being the liberation of several prisoners on Level 6 to augment the strength of his crew), Crocodile was dismissed from his position after his defeat (although he was succeeded by Blackbeard), and Gekko Moriah was deemed "too weak" to retain his status. Furthermore, both Monkey D. Garp and Sengoku resigned after the Battle of Marineford.

During Timeskip
During the timeskip, all three groups underwent changes in membership. In the Marines, Admirals Akainu and Aokiji dueled for the right to become Sengoku's successor. Sakazuki won and has been promoted from Admiral to Fleet Admiral, while Kuzan resigned from his position as Admiral. Of the Shichibukai, Boa Hancock, Bartholomew Kuma, Donquixote Doflamingo, and Dracule Mihawk retained their titles, and Trafalgar Law, Buggy, and Edward Weevil were appointed to fill the three vacant positions. Of the Yonko, Marco took command of the Whitebeard Pirates, but is not known to have taken his former captain's title of Yonko. A year after Whitebeard's death, the Whitebeard Pirates and Blackbeard Pirates clashed in the Payback War. The Blackbeard Pirates ultimately emerged victorious, and the Whitebeard Pirates have not been seen since. This event, combined with the Blackbeard Pirates conquering the majority of Whitebeard's former territories, led to the World Government acknowledging Blackbeard as a Yonko.

Dressrosa Saga
After the timeskip, Law's status as a Shichibukai was ultimately revoked due to his forming an alliance with the Straw Hat Pirates, and Doflamingo also lost his status after the Dressrosa incident and the discovery of his abuse of power.