Tobi

also known as in Akatsuki, was formerly Zetsu's subordinate. Madara became a full Akatsuki member after Sasori's death and afterwards wore the same ring as Sasori having recovered it after his death and taking his place as Deidara's partner. After Deidara's death, he revealed his real name and that he had been the one giving Pein orders, making him the true leader of Akatsuki.

Background
Madara Uchiha is the founder of the Uchiha clan. For reasons unknown, Madara's relation with Konohagakure, the Uchiha clan's home village, eventually became disrupted, and Madara left the village. The First Hokage, however, chased after Madara, and fought him at the Valley of the End, the battle ending with Madara's defeat. To commemorate the battle, two statues, of Madara and the First Hokage, were carved into the earth around the waterfall.

While this has not been verified, Jiraiya suspects Madara had some connection to the Nine-Tailed Demon Fox's attack on Konohagakure many years after Madara's death. The demon fox himself seems to have a connection to both Madara and the Uchiha clan, commenting to Sasuke Uchiha that his chakra is just as evil as Madara's.

Madara also seems to have a connection to the founding of the organization Akatsuki. While the organization's founding has not yet been detailed, it can be presumed Madara founded Akatsuki and placed Pein as the figure-head leader, with Madara giving Pein orders from the background. Outside of Pein, the only members who are shown knowing Madara's position as the true leader are Konan and Zetsu.

Personality
Madara is very calm and serious, a trait he shares with most of the Akatsuki. He speaks with less formal diction, such as the using honorifics, and shows typically 'villainous' arrogance. Dispite this apparent arrogance, he is not above complimenting the the abilities of Sasuke Uchiha and Naruto Uzumaki, if only for the interest of Sasuke's Sharingan and Naruto's tailed beast.

Tobi
For whatever reasons, Madara eventually took on the guise of and was placed within Akatsuki as Zetsu's subordinate. As Tobi, Madara wears an orange mask the covers his entire face outside of his right eye. Furthermore, Tobi has several bolts or pins present on the arms of his uniform, obscured by his Akatsuki cloak. While acting as Tobi, unlike others within Akatsuki, Madara is carefree and goofy, and is also implied by Zetsu to be relatively young. While Tobi, Madara also uses a very formal and correct manner of speech.

While Tobi greatly respects his other members, calling Deidara Deidara-senpai, most of the other members tend to get easily annoyed by Tobi. Deidara, who believes all within Akatsuki should be serious and calm, is not pleased with Tobi's childish personality, and frequently attacks Tobi in a comedic fashion when annoyed. Kisame, on the other hand, somewhat appreciates Tobi's ability to brighten up a gloomy organization such as theirs.

As Tobi, Madara has not yet been seen wearing a forehead protector, though his Uchiha origins suggest his home village is Konohagakure.

Abilities
Madara's area of expertise or specific powers remain unclear, as he has never been shown fighting or utilizing jutsu, although since the Uchiha clan is known for its fire techniques it is possible he knows at least basic fire jutsus. He and Deidara were able to defeat the Three Tails, but Madara (as Tobi) was never shown doing anything but running from it. After the battle, which goes largely unwitnessed, he claims to have knocked it out with a single attack. Deidara disagrees with the assessment, claiming that it was his "artistic contribution" (exploding clay) that allowed them to win. Madara (as Tobi) is also shown planting Deidara's clay minefields underground during a later battle.

Madara shows the ability to recover from what are presumably debilitating or fatal blows. After being slashed across the midsection by Sasuke's sword, Madara (as Tobi) collapses, only to stand up again a few seconds later and complain about the speed of the attack, and later when Naruto hits him in the back with a Rasengan. The attacks literally phase through him, avoiding damage.

Madara also shows the ability to travel long distances within relatively short timespans, escaping the final explosion created by Deidara and reappear in Amegakure, then travel to the Valley of the End all within a very short time span.

After he escaped Deidara's final explosion, he revealed he had the Sharingan, one visible through the opening of his mask. The demon fox mentioned that Madara Uchiha has more sinister chakra than his own. According to Jiraiya, Madara was the only person capable of summoning the demon fox, which was said to only come where human malice collects and festers.

Rescue Gaara arc
Madara is first introduced as Tobi alongside Zetsu collecting the rings of Deidara and Sasori. While Sasori's ring is collected, Deidara, who managed to live, appears and reclaims his own ring. After annoying Deidara three times, Deidara responds by comically choking Tobi with his feet. Using Sasori's ring, Tobi manages to join Akatsuki, though Zetsu claims isn't as easy as collecting the ring.

Hidan & Kakuzu arc
Tobi, who by now has successfully joined Akatsuki, is partnered with Deidara and sent to collect the Three-Tailed Beast. After Deidara reveals that this is Tobi's mission, Tobi, panicking, is seen only running away from the Three-Tails in a comic fashion. After the battle, which goes mostly unwitnessed, Tobi triumphantly claims to have defeated Three-Tails in a single blow, though Deidara disagrees. Tobi later falls asleep atop the Three-Tails, much to the anger of Deidara, who "wakes him up" by detonating clay next to him.

Sasuke vs. Deidara arc
Tobi is seen following Deidara, who is chasing Sasuke Uchiha, and eventually join Deidara's battle with Sasuke, though he does little besides laying mines for him and serving as a distraction. When Deidara prepares to use C4, Deidara orders Tobi to flee, though, while escaping C4, Tobi is later shown getting killed by Deidara's final explosion.

It is later revealed, however, that Madara lived, as he is seen ordering Pein in Amegakure just a few minutes after Deidara's death. Madara gives Pein the order to personally capture Naruto Uzumaki. Madara then travels to the Valley of the End, and while resting there comments that Sasuke Uchiha's Sharingan is growing more powerful.

Itachi vs. Sasuke arc
Madara appears as Tobi to stop Naruto, Hinata, Yamato, Kakashi, Sakura, Shino, Sai, and Kiba from searching for Sasuke and Itachi Uchiha, and phases through all attacks the group sends, even a full-powered Rasengan from Naruto Uzumaki.

Past Speculation
Before Tobi is revealed to be Madara Uchiha, several fans hypothesized that he could be Obito Uchiha, as his only shown eye was the opposite of the eye that was inserted in Kakashi, which replace Kakashi's destroyed left eye in the manga-exclusive Kakashi Gaiden, and their similar hair styles. The spelling of Obito was also similar to Tobi, creating the fan reference of ObiTobi, or Tobito.

Another fan speculation is that Pein, who now is discovered to have multiple bodies, could have used the remains of Obito to reconstruct the body and revive Madara in the body of Obito, which contained Sharingan only within the right eye, the same eye that Madara has been shown using the Sharingan. It is also possible that Pein employed Kakuzu, who was able to reattach Deidara's arms after they were severed, and could have done similarily with Obito and Madara.

There are also speculations that Madara Uchiha is the third Uchiha clan member with the ability to possess the Mangekyo Sharingan as mentioned by Itachi Uchiha.

One more speculation is that Orochimaru, who was able to use the Summoning: Impure World Resurrection, as demonstrated in the fight with the Third hokage, and who was once a member of Akatsuki, probably used or learned it there. And so, it is possible that the body of Obito has been reanimated and the spirit of Madara has been implanted into him using another forbidden jutsu.