He is in fact the the biological son of Gozaburo Kaiba, the adoptive brother of Seto and Mokuba Kaiba, whose mind was digitalized by his father after his death (vegetative coma in the Dub). He yearns to take over Seto's body, both to is take his place as CEO of his father's Kaiba Corp multinational, and to take his revenge.
In the original Japanese, he is the mastermind behind the plot and only supplanted by his father after his defeat, but in the dub he was his father's pawn all along. In both, he ends up reforming and sacrifices himself to save everyone as an atonement.
He was voiced by Andrew Rannells in the anime.
Noah Kaiba was born about the same time as Seto. His father, the powerful weapon mogul Gozaburo Kaiba hired many servants and tutor to raise him in luxury, catering to his every need and teaching him intensely in many subjects, from the academic to violin playing and horseriding.
Everything ended the fated day in which Noah was killed in a car accident. (The dub merely states that his body was rendered useless, suggesting a vegetative coma). Desperate to keep his heir alive, Gozaburo uploaded his mind into a virtual cyberspace on a supercomputer. He then adopted Seto Kaiba to have Noah take over his body, but soon forgot about him and concentrated his efforts on making Seto his heir.
Noah was aware of it and he deeply resented his adoptive brothers for stealing the attention of his father, while he was trapped in a false reality where all the virtual people devised to "keep him company" always acted the same way no matter what he did.
In a desperate attempt to regain his father's attention, he vastly improved the virtual world and assumed complete control of it, absorbing immense amount of data in the process, and devised extremely powerful weaponry and missile satellite for his father to sell. Yet, this attempt barely earned Gozaburo's acknowledgement for a short while.
Contrary to Noah, Seto and Mokuba were treated awfully, being heaped lesson after lesson without free time in what amounted to psychological torture and physical punishments. As such, they plotted to take over the society and successfully did, turning it into the biggest game designing company in the world.
Six years later, Noah decided to take revenge on Seto. He recruited the Big Five (who had become trapped in a previous virtual world, as their bodies in the outside world were destroyed), promising to get them new bodies to use if they helped him get his revenge and take over the Kaiba Corp.
After the first half of Battle City final, as Seto's aircraft was flying towards the Duel Tower, that Seto had built over the ruins of the Kaiba Corp's former main factory, Noah took control of the ship and directed it to his submarine fortress, before ordering almost anyone inside it to enter the Virtual World.
There, he staged several duels between the Big Five and the protagonists, while putting Seto and Mokuba into a virtual reconstitution of their childhood and setting some hints about his identity all along the way.
Eventually, every member of the Big Five got defeated, and one of them even kidnapped Mokuba whom Noah took back and brought into his own virtual lounge. Irked by both the Big Five's cheating and Mokuba's unfaltering support of Seto, Noah cast his former associates into remote areas of the virtual world where they would remain trapped forever, before brainwashing Mokuba and making him believe that he was his brother and that Seto was a malevolent stranger.
Noah then took the matter in hand and challenged Seto into a duel at the very core of the Virtual World. Noah proved more than a match even for Seto himself, but as he was (seemingly) on the verge of losing, he used a brainwashed Mokuba as a shield, forcing Seto to cancel his attack and to lose the duel. (It is worth nothing to note that Kaiba's final attack would not have defeated him in the first place, as he had a way to replenish his Life Points should he take the hit and waited until the last moment to activate it).
Noah then turned the two brothers into stone, but an outraged Yugi Muto took over the duel, adding Seto's remaining cards to his own and starting the duel with the number of Life Points that Seto had left when he lost. In spite of a huge gap in Life Points that continually increased as the duel progressed, and the fact that Noah turned one of his friends into stone in each turn, Yugi managed to summon Seto's three Blue-Eyes White Dragons and used them to defeat Noah once and for all.
Gozaburo reveals himself
Gozaburo then appeared and revealed that when Seto took over the Kaiba Corp from him, he secretly uploaded his own mind into the supercomputer. He takes profit in Noah's defeat to hijack control of the virtual world; leaving Noah as surprised as the others by Gozaburo's appearance. (The 4Kids dub changed it by making Gozaburo as the one pulling the strings all along, with Noah knowingly working for him).
Now that he has been defeated, Gozaburo bluntly tells him that he no longer needs him, calling him a failure and a disgrace. This explains further why Noah would ultimately choose to turn on his father (Noah tried all his life and afterlife to be acknowledged by his father, only to learn that he was but merely as an instrument).
Distraught, Noah then pretended to lead the protagonists outside of the virtual world, only to isolate Mokuba from the others, whom he dispersed away in the virtual world. Initially indifferent to Mokuba's efforts to cheer him up, he took over Mokuba's body, and returned to the real world at last, before setting the submarine fortress to explode (which would delete the virtual world and everyone in it).
However, remembering Mokuba's undeserved kindness towards him despite everything he put him through caused a change of heart. (Mokuba kept calling him brother and empathized with Gozaburo's abandonment and rejection of his son, no different that the abuse he and his brother suffered at his hands.)
Unable to stop the missile despite his best efforts, Noah returned to the virtual world to help everyone escape. He held Gozaburo (who had lost a duel against Seto and had turned into a gigantic beast of flame, in a last-ditch attempt to destroy his adopted son once and for all) in place as the missile destroyed his submarine fortress and the supercomputer hosting their minds, deleting the Big Five and Gozaburo. Having finally come to term with the fact that he was dead and no longer belonged in this world, Noah bid a tearful goodbye to Mokuba, thanking him for his kindness and welcomed death as a respite from his years as a virtual ghost.
The 4Kids dub censored Noah's death, having Yugi and the others suspecting that Noah uploaded his mind on a backup drive. It is unknown whether or not he did, and he has never been seen since the destruction of the virtual world, except for some flashbacks.
Noah is collected, polite, and quite respectful to those he does not bear grudge against, speaking in a formal, business-like tone; though he is arrogant and smug, having a creepy un-childlike behavior. The 4Kids dub drops all poise and politeness and makes him extremely conceited, obnoxious and taunting.
He is as intelligent as Seto, as evidenced by his ability to devise highly elaborate weaponry and machinery, or to hack his way into any kind of computer. Since his mind is linked to the virtual world, he always knows everything that happens in it.
Due to having been extremely sheltered and doted upon as a child, he is pretty immature, being used to having all his whims answered and disliking when things do not go his way. Seto notes that being stuck as a child for all these years, he never had a chance to mature out of his spoiled demeanor, hence his childish tantrums and obsession with always being in charge.
Although he boasts about his god-like control over the virtual world, stating that all of the information he got from computers all over the word that made him transcend humanity; it is pretty clear that he despises the shallow travesty of reality he is stuck in and wants to escape to the real world no matter the cost.
It is made quite clear that behind his arrogance and hatred, lies a scared child desperate to return to any semblance of normalcy, lashing out years of pent-up frustration from his gilded prison at his step-brothers.
Noah regards himself as the rightful heir of the Kaiba Corp, and treats Seto and Mokuba with contempt and jealousy. However, the reason of his hatred for them is that they took the place that was his own while he was left in a hapless state, barred from both a normal life and from the Afterlife. He holds them responsible for his father's loss of interest towards him, although in reality the three of them were nothing but assets to use or discard in Gozaburo's eyes.
Noah looks up to his father and craves for his acknowledgement, desperately trying to prove his worth as a heir. He believes himself superior to his step-brothers and strives to prove it to them and to the world by defeating them, and envies their extremely close sibling bond; doing all that he can to destroy it and going as far as brainwashing Mokuba to take it for him.
The pain of being abandoned by his father and the loneliness of his years in the virtual world being the motivation for all he does. It is the distress of being called worthless by his father that first makes him rage quit the Virtual World and set its destruction, then to turn his back on Gozaburo.
In the end, he is appreciative and thankful to Mokuba, the only one to ever show him genuine, disinterested kindness due to seeing him as another victim of Gozaburo's selfishness.
Noah appears to disapprove of cheating, as he nearly pulled the plug on Chikuzen Ooka of the Big Five for cheating during his duel with Katsuya Jonouchi, but it seems that what really irks him is when people disrespect the rules he set, since he himself has no qualms into manipulating Mokuba to interfere with his duel. However, given that he was not really in danger of losing and that he dueled fairly against Yugi afterwards, he might have done it only to toy with Seto and Mokuba's sibling bond.
Noah is an immensely talented duelist in his own right, leagues above any of the Big Five and even his father, more than able to hold his own against Yugi and Seto themselves. In fact he is one of the only two villains who pushed Yugi into such a corner that he considered forfeiting (with the other one being Dartz). (It's worth noting that they are both filler villains).
Noah uses two different types of Decks, one based on the creation of the world which mixes cards based on the prehistory, the dinosaurs and even science fiction; the other based on the (very hard to play) Spirit Monsters, who return to his hand at the end of his turn, making them almost undefeatable but forcing him to summon them at every turn and greatly jeopardizing his defenses.
Noah relies mostly on raising his Life Points at every turn to reach ludicrous amounts, mostly through a card that increases them whenever a Spirit Monster returns to his hand and through his Ace Card (literally) the Ark of Miracles, while at the same time turning his opponent's moves to his advantage and summoning powerful monsters which he grants additional abilities.