Root A is the anime second season.
The bit from the manga that Root A replaces doesn't really have a name of its own to my knowledge.
Major plot changes:
Rather than join Aogiri and do a bunch of attacks on the CCG, break into Cochlea, then return to Anteiku for the final battle, where Hide dies of a wound and Kaneki carries his body to the CCG in a long ass walk, this happens:
SPOILER ALERT IN CASE ANYONE DOESN'T GET IT
Kaneki doesn't join Aogiri, though he still leaves Anteiku. Alongside Tsukiyama, Banjou (and his two friends) and Hinami, he begins investigating Kanou, the doctor who turned him into a half ghoul. The main sequence of events here (other stuff happens, like investigation into Rize and the one eyed king and a fight with Aogiri) includes taking down the ghoul restaurant to find Madam A, a ghoul connected to Kanou, who would later lead Kaneki to Kanou's mansion. Kanou tries to convince Kaneki to join him in the "salvation" of humanity through ghoulification. Also, Rize is revealed to still be alive, and Kanou is harvesting her regenerating organs for the ghoulification process, though Yomo shows up and steals Rize. Then the CCG shows up. At this point, Kaneki canabalises Kanou's failed ghoulification experiments and develops his Kakuja. Here, he has his fight with Arata Shinohara, Juuzou takes on the half ghoul girls with his knives and scythe, and eventually Kaneki escapes the mansion and the CCG.
Some time later, Kaneki is all depressed from going insane in his Kakuja state and almost killing Banjou. Hinami convinces him to come to an author book signing to cheer him up. Surprise surprise, it's Sen Takatsuki's book signing. Kaneki heads over to Anteiku and has his conversation with the manager about the one eyed owl and his daughter Eto.
Then the owl suppression stuff happens. Most of this plays out the same as the anime's finale, with Anteiku being attacked, Yoshimura fighting in his Kakuja and eventually being defeated, the Dog ghoul and Monkey ghoul fighting the CCG, then Kaneki shows up and saves them. He tells them to leave through the underground tunnel V14 to escape. The one eyed owl appears, fights with Shinohara and Juuzou, cutting off Juuzou's leg and incapacitating Shinohara. Kaneki has his fight with Amon, where he chops off Amon's arm, while also having most of his torso destroyed. Kaneki flees into the sewer system. Here, he starts going insane because of his Kakuja again, and Hide shows up. Hide reveals he knew that Kaneki had become a ghoul all along, and is still going to help him, telling him that he needs to go all out one more time. Kaneki blacks out. When he wakes up, Hide is gone, there is "a sweet taste" in Kaneki's mouth, and it is implied that Kaneki ate Hide.
Kaneki makes his way through V14, and finds Kishou Arima standing there amidst a mountain of ghoul corpses who tried to escape that way
"No ghoul may pass V14".
It is implied that the members of Anteiku who went that way also were killed. Kaneki fights with Arima. However, Arima defeats him,
stabbing him in both eyes (because Kaneki kept fighting after only being stabbed in one)
through his head. During the fight, Kaneki manages to crack IXA and scratch Arima's cheek, being the only ghoul who ever did so.
Kaneki has one of his dream state things, where he has a nice chat with his previous non-white-haired self and reflects on the events of the manga. At the very end, he has a flashback to the night the steel beams crushed Rize, and remembers something he never saw before - at the top of the building from which the beams fell, there was a ghoul in a clown mask.
Back in the real world, the One-Eyed Owl is still causing chaos. Aogiri Tree has shown up. Amon wakes up from his fight with Kaneki and meets with Seidou Takizawa (the boy who was Akira Mado's rival), where they meet with Tatara and Noro. Noro eats Takizawa's limbs, Amon begins to fight Tatara, but the fight is not shown.
Like in the anime, the CCG is getting slaughtered by the one-eyed owl, but Arima shows up and forces it to retreat. The owl grabs Yoshimura on the way, and later reveals her identity as Eto / Sen.
Aftermath is:
- 99% of ghouls in the area killed
- Kaneki is missing, to both ghouls and the CCG
- Tsukiyama is starving himself out of grief at Kaneki
- Kanou has gained a supply of ghouls (including Yoshimura) and ghoul investigators to continue his experiments
- Arima is given "ownership and identity rights" of a new "quinque" and Kaneki's has become an unperso, with his records being expunged.
Itori, Roma (the clumsy staff girl at Anteiku) and Uta are revealed to be a group called The Clowns, who basically cause all sorts of nasty stuff to happen then sit back and watch for ***** and giggles.
Touka is all sad that Kaneki is gone. She monologues and concludes that she believes Kaneki will return one day.
Original manga ends here.:Re Starts here
Edit: Oh and there's a two page epilogue chapter that's just a lead up to :re, where Akira visits her father's grave, says the "chimera quinque" project has been approved, and that she's been assigned alongside Arima to a new division, with a "certain junior" who is "different from the others". At that moment, said junior shows up and introduces himself as Haise Sasaki.