There have been many groups of malicious people in Naruto. The Seven Deadly Swordsman, Kara, Taka, The Sound Five... none are as iconic as the Akatsuki, though. There have been different iterations of the Akatsuki with different motives, but their beliefs are typically similar.
The first order of the Akatsuki was founded by Yahiko to protect the Village Hidden in the Rain. The second order was made by Nagato to carry on Madara's plan to fulfill the prophecy written on the Uchiha Stone, and Obito's third order of the Akatsuki was a continuation of Nagato's. The fourth order of the Akatsuki was made by Shin Uchiha, and it seemed to only hold the purpose of keeping the Akatsuki moving. Let's take a look at the group's most powerful members. This list will only include Akatsuki members as they were in the Akatsuki, not necessarily in their primes.
Sasori is a puppet master who was both a puppet maker and an assassin before joining the Akatsuki. His most impressive assassination was killing the Third Kazekage, and afterward, he made him into a powerful puppet, fully capable of wielding the Iron Sand. He's made many people into puppets over the years, including his own parents and even himself.
Because he made himself into a puppet when he was young, he holds the guise of a young boy. Because he has expanded his parameters of chakra control beyond that of any regular ninja, he is capable of wielding a hundred puppets at a time instead of just one for each finger. All of his weapons are imbued with poison, and he's known to be able to attack with millions of microscopic homing projectiles of Iron Sand at the same time.
9 Itachi Uchiha
Itachi is Sasuke Uchiha's older brother, and he had assassinated his whole clan save his brother long before joining the Akatsuki. He was a member of the Anbu as an assassin alongside another Uchiha member named Shisui, who was killed by Danzo. Danzo gave the order for the Uchiha clan to be wiped out, and Itachi was the one to carry out the deed.
Itachi has some of the most formidable ocular jutsu of the Akatsuki, capable of using the Tsukiyomi, which can create a genjutsu world in which he can control all of matter, space, and time, creating infinite copies of himself. He can use the Susano'o technique, in which he can seal people away with it, and he can control fate itself, using the Izanagi and the Izanami.
Pain is the animated corpse of Yahiko, the founder of the original Akatsuki. He serves as the leader of the second order of the Akatsuki, though he is actually being remotely controlled by Nagato Uzumaki.
Pain possesses a projection of Nagato's Rinnegan that he inherited from Madara, which gives him access to a wide variety of abilities. He can control gravity using the Shinra Tensei and the Chibaku Tensei, and he is capable of using the Six Paths, which allow him to summon powerful animals, summon the God of Death, pull out people's souls, turn his body cybernetic, and several other techniques. He can also resurrect people he has recently killed.
Kakuzu joined the Akatsuki in hopes to expand his fortune, and he's known to be one of the greediest people in the Shinobi World. He is known to have actually taken out multiple Jinchuriki, including Fuu and Yugito, and he has slain entire villages alone without any help from Hidan.
Kakuzu managed to successfully make himself immortal by making deals with demons by letting them into his body and attaining five hearts, each of which are represented on his body as masks that use different chakra natures. He is so durable that Matatabi and Kakashi are incapable of damaging him under normal circumstances.
Kisame is a man holds many qualities of a shark, having been trained by one of the Seven Deadly Swordsman, the former wielder of the Samehada. After his master killed himself, he took on the blade. He tamed this sentient sword and learned to merge with it upon will, being in full control after the merger.
Kisame is known as the "Tailless Tailed Beast", noted to have both the chakra reserves and power of a Tailed Beast. He fights by draining the chakra of his opponents constantly during battle, and he has fought against Killer B. In his second encounter with Killer B, he managed to drain most of his chakra and make his way back to the Akatsuki with intel.
Zetsu is undoubtedly one of the most intelligent characters in all of Naruto. Being one of Kaguya's "sons", he planned for millennia to bring her back. He actually wrote the prophecies on the Uchiha Stone, to set in motion the events that lead to Madara passing on his wishes to Nagato so he can create the second order of the Akatsuki.
A bunch of weaker Zetsu clones were made en masse that are capable of covering a lot of ground and killing a lot of people. Black Zetsu is capable of leeching onto powerful people, to the extent that Madara was defeated once and for all because Zetsu latched onto him and caught him off guard with a finishing blow.
Orochimaru has been all over the Shinobi World, and he is perhaps the most knowledgeable ninja since Hagoromo regarding jutsu. His incentive to join the Akatsuki was to study the jutsu that the people of the Akatsuki use, and his ultimate goal is to become truly immortal and to master every possible jutsu.
One of the most formidable things about Orochimaru is that it's hard to permanently put him down. Multiple users of genjutsu have successfully caught Orochimaru into their techniques, and Orochimaru always comes back. Orochimaru was sealed away by Itachi, but he always comes back. Orochimaru can also use the Edo Tensei technique to summon Hashirama and Tobirama and Minato to fight against potential opponents. He also has a universe he created inside of his mind that he uses to take other people's bodies.
3 Obito Uchiha
Obito is the founder of the third order of the Akatsuki, having been a disciple of Minato, the Fourth Hokage. While he was presumed to have died many years ago as a child, he has secretly watched Kakashi during his missions and helped him out in times of need without getting caught. He was eventually the cause of Minato's death.
Obito has genjutsu even greater than the Kotoamatsukami, being capable of keeping Yagura under his control for many years without Ao noticing, despite immediately noticing the usage of Kotoamatsukami. Obito later inherited the Rinnegan previously used by Nagato, mastering it. While using it, the Six Paths he controlled were all fully-functional Jinchuriki.
2 Madara Uchiha
Madara was the first notable person that Zetsu involved in his plan to bring back Kaguya, and he was with him throughout the entirety of it. Madara is also capable of using the Tsukiyomi, the Izanagi, and the Izanami, and with his Rinnegan, he can perform all of the abilities Nagato and Obito could. He also uses it to perform Limbo, which lets him fight from other universal planes of existence.
Madara has developed his Susano'o further, perfecting it until it became destruction incarnate with the power to smash everything in the world. He defeated the Five Kage in office at the time of the Fourth Great Ninja War, even while holding back severely. He eventually became the Jinchuriki to the Ten-Tailed Beast, gaining power that was said to be nearing Hagoromo's, who is practically the God of Shinobi who created the moon.
1 Shin Uchiha
Shin Uchiha was an experiment created by Orochimaru during the events of the Boruto manga. He took inspiration from Danzo and imbued a single artificial life form with as many Sharingan as he could get his hands on. Shin eventually fully awakened his Sharingan, and then he cloned himself repeatedly until he recreated the whole Uchiha clan.
Shin was a threat to Naruto and Sasuke. At this time, even base Naruto and Sasuke are known to be able to single-handedly defeat everyone in the Shinobi World by themselves, and Shin Uchiha is also stated to be a threat to the whole world. Naruto and Sasuke also took on Momoshiki shortly after, who created a parallel universe with his own power, and who was feared immensely by Kaguya. On sheer statistics alone, he far outclasses the rest of the Akatsuki.