The Akatsuki is a group of individuals who work outside their village policies. They're seen wearing black coats with red clouds and their headbands slashed, which shows that they have been disowned by their villages. The main goal of the Akatsuki is to secure peace and make the world a better place. They have different hideouts which are inaccessible to anyone outside the team.
(Please be aware that there are spoilers in this article)
Yahiko was the founder of the Akatsuki along with Nagato and Konan during the Third Great Shinobi War. They created Akatsuki to bring peace to their motherland which always got caught between wars. Tobi (at the time who referred to himself as Madara Uchiha) approached them and offered to teach Nagato how to control the Rinnegan and mentioned how Nagato was the reincarnation of the Sage of Six Paths.
The Akatsuki's objective to end war saw a lot of Ame ninjas join them and help them their cause. Due to Yahiko's natural charisma and leadership skills, they soon gained a name for themselves. As soon as Hanzo was aware of the Akatsuki, he reached an agreement with Danzo, of the Hidden Leaf Village, to get rid of them.
When Yahiko, Nagato, and Konan meet Hanzo in order to find common ground and reach a peace agreement, he ambushed them with both his and Danzo's men, and takes Konan hostage. This forces Yahiko to commit suicide to save his friend which enraged Nagato and he summoned the Demonic Statue of the other path and used it to slaughter Hanzo's men but Hanzo himself escaped.
Yahiko then entrusted the goal of attaining peace to Nagato and after Yahiko's death, Nagato realized that Akatsuki's pacifist approach was a failure in attaining peace. Instead he decided that they would make the world experience such death and destruction that it would never think of war ever again.
From that day onwards Nagato started using the alias 'Pain' and after the Third Great Shinobi War, they began recruiting powerful and talented shinobi. In return for the shinobi’s services, they were promised protection and a chance to can fulfill their objectives. After recruiting 10 members, the Akatsuki made objectives to reach their goals:
The main objective behind catching the tailed beasts was to harness their Tailed Beast chakra and use it to wipe out the conflicts in one go itself, which would frighten the world and make them stop the war.
The Akatsuki consisted of 10 members who were divided into a team of two people. They were paired up based on similarities or their combination was compatible. The teams of the Akatsuki were:
As they were childhood friends and the founding members of the Akatsuki, Nagato and Konan were a team. They got along very well too. As Nagato was the leader, Konan did whatever he instructed her, even though there were times when she disagreed. Nagato's real identity was hidden from the organization and only Konan knew his true form. He died in Konoha, after sacrificing himself to revive everyone in the village after it had been destroyed. Konan also dies later on, while fighting Tobi when he came to steal Nagato's Rinnegan from his dead body.
Black and White Zetsu joined the Akatsuki way before Tobi and helped Tobi approach Nagato. Although they're two different individuals, the majority of the time they're seen merged and act as an singular unit. They sometimes argue but they don't let it hinder their duties or missions at any cost. They help transport the jinchuriki to the Akatsuki hideout in order to extract the Tailed Beasts from them. During the Fourth Great Shinobi War, Black Zetsu is sealed by Naruto Uzumaki while White Zetsu is killed by Sasuke Uchiha.
Orochimaru and Sasori were one of the first teams to be formed in the Akatsuki. They were very efficient and compatible, and even got along well to the point where Orochimaru was aware of what Sasori's real body looked like beneath his cape.
Sasori was quite annoyed with Orochimaru when he started distancing him from Akatsuki’s activities and took a personal interest in his own activities. This deteriorated their partnership which led them to change their partners. Orochimaru was killer by Sasuke Uchiha while Sasori was killed by Lady Chiho and Sakura.
Itachi and Juzo also worked very well as they had their own unique system of formations which synchronized their attacks with each other. They would use Itachi's genjutsu as an opening attack while Juzo incapacitated the person with his sword. Juzo was killed during the assassination of the Fourth Mizukage.
After the death of his partner, Itachi was partnered with Kisame. Their partnership was appropriate as they were both known for their notorious killing sprees. However, despite the fact that they both had conflicting personalities, they got along well. Even though Kisame was quite violent he always followed Itachi's instructions. Not only that, he liked the Uchiha, so much so that he was quite sad when Itachi died in his battle with Sasuke. Kisame was later killed by Might Guy and his team.
Deidara replaced Orochimaru and is paired up with Sasori forcibly. As they both view themselves as artists, they always find the other one inferior to theirs. Sasori believes that art is eternal whereas Deidara believes that art is an explosion and flashy. Despite their conflicting views, he did pay some respect to Sasori because he was more experienced but was very disappointed by Sasori's death.
As Kakuzu used to kill his teammates in a form of rage and steal their hearts, he was paired with Hidan, who was an immortal. All the other teams had cooperation or even had tiny moments of cooperation but it was quite the opposite for Hidan and Kakuzu. They both actively hated each other as Kakuzu's obsession with money annoys Hidan and Kakuzu wants to kill Hidan because of his annoying habits. Even though they hate each other they're known as the most effective team in the Akatsuki. Hidan was buried deep down by Shikamaru while Kakuzu was killed by Kakashi Hatake.
After the death of Sasori, Tobi was paired up with Deidara. As it is seen Tobi annoys Deidara, which makes him think that Sasori was a better partner. He annoys him to a point where Deidara feels like killing him. However, Deidara is unaware of Tobi's true identity and takes him on as a disciple and tells him about the ways of the world and how to succeed in the Akatsuki. Deidara attempts suicide while fighting Sasuke Uchiha and dies.
A. The red clouds represent the rain of blood that fell on Amegakure during the time of war.
A. According to the fan votes, Itachi Uchiha was the most liked member of the Akatsuki.
A. Zetsu was considered the weakest member of the Akatsuki because of his fighting abilities.