Noah Kaiba is the biological son of Gozaburo Kaiba, and the central antagonist of the anime exclusive Virtual World Story Arc: of the first half of Season 3, Enter the Shadow Realm, in Yu-Gi-Oh!. Despite being the main antagonist of the third Story Arc, he was in fact a pawn (unwitting in the original Japanese version) to the season's true main villain.
History
Past
Noah was killed in a car accident, so 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, as the Kaiba Corp was at the time manufacturing weapons. Yet, this attempt barely earned Gozaburo's acknowledgement for a short while.
Noah's Revenge
6 years later, Noah decided to take revenge on Seto. He recruited the Big Five (who had become trapped in the 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, Noahh 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 cheatings and Mokuba's unfaltering support of Seto, Noah cast his former associates into a remote area 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 on the verge of losing, he used 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 an effect left to replenish his Life Points should he take the hit).
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 Noahh 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 Noahh once and for all.
Gozaburo reveals himself
Gozaburo then appeared and revealed that when Seto took over the Kaiba Corp from him, he too uploaded his own mind into the virtual world. He takes profit in Noah's defeat to hijack the world; and Noah is as surprised as the others by Gozaburo's appearance (the 4Kids dub changed it by making Gozaburo as the one pulling the strings, with Noah working for him all along).
Now that he has been defeated, Gozaburo 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, he eventually had a change of heart and 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 virtual world was destroyed, Noah and Gozaburo along with it.
It is suspected by Yugi and the others that Noah saved 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. This speculation only occurred in the 4Kids dub to censor his death, as he is the only one of the seven villains killed in the virtual world to turn good.
Personality
Personality-wise, Noah is polite and collected (extremely conceited and obnoxious in the 4Kids version). Due to having been extremely sheltered and doted upon as a child, he is used to having all his whims answered and dislikes 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.
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 barred from both a normal life and the afterlife. He believes that his father preferred them over him, although in reality the three of them were nothing but assets to use or discard in Gozaburo's eyes.
Noah 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. Although he boasts about his god-like control over the virtual world, stating that all the information he got from computers all over the word that made him transcend humanity; it is pretty clear that he despises the travesty of reality he is stuck in and wants to escape to the real world no matter the cost.
Noah looks up to his father and craves for his acknowledgement, desperately trying to prove his worth as a heir. He wants to prove superior to his step-brothers 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 expresses gratitude to Mokuba, the only one to ever show him genuine 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.
Deck
Noah is a highly talented duelist in his own right, leagues above any of the Big Five and far better than 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 nothing to note that they are both filler villains).
Noah with Shinato behind
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.