Note: "Kaiba" references Seto Kaiba. Any other family members will be signified, i.e. "Gozaburo Kaiba".
Seto has a little brother named Mokuba. They were both orphans until Gozaburo Kaiba decided he needed an heir, someone to inherit Kaiba Corp when he is gone. He chose Seto because he was an intelligent individual (he often played chess with him to test him). Gozaburo didn't plan on giving Kaiba Corp to Seto right away, but later, at a time of his choosing. Seto, however, took initiative and came up with a plan to take over Kaiba Corp. Gozaburo owned 51% of Kaiba Corp (he actually owned 49% 2% that his advisors, the Big Five owned) Seto owned 48%, and Mokuba owned 1%. Mokuba , of course, agreed to give his 1% to his big brother. And when Seto persuaded The Five to give him their 2%, he owned 51% of Kaiba Corp, chasing his step-father, Gozaburo, out.
Kaiba later fired The Five, which later, they try to get revenge on him. Gozaburo will later digitalize himself into a virtual world he created and, only much later, try to take Kaiba's body. It is on this occasion that The Five will try to get their revenge on Kaiba (and will fail at the hands of Yugi, Joey, Tea, Duke and Kaiba himself). Gozaburo is ultimately unsuccessful, and, eventually, destroyed along with the world he created.
Kaiba Corp was a weapon factory during the time of Gozaburo Kaiba. When Seto Kaiba takes over, however, it becomes focused on the "Duel Monsters" card games, creating different ways to improve the game. In fact, Kaiba Corp created the Duel domes and all three versions of the Duel-Discs.
Kaiba is extremely arrogant and self-centered with little time for anybody beyond himself and his younger brother Mokuba, making him a typical anti-hero. He is dismissive in his manner of speech represented in the original Japanese version of the series by his omission of honorifics, and his use of the boastful word ore to mean "I" instead of the usual boku.
Although magical powers and items are often employed in the series, as a rational man, Kaiba does not believe in such things and, when necessary, convinces himself that it is all tricks and illusion. Although this trend had a tendency to be exaggerated in the English version of the series, it is present in both language versions of the anime and manga. Eventually he grudgingly comes to accept that there is little explanation for the bizarre, fantastical events he witnesses throughout the series. In a similar vein, Kaiba is shown to have an affinity for machines and as such manufactures several holographic devices for Duel Monsters and, as suggested by Anzu Mazaki (Tea Gardner) in the English dubbed anime, this may be the cause of his misanthropy.
Kaiba changes in personality in the second season of the second series anime onwards. In the first season, Kaiba acknowledges Pegasus' ability to read minds, as well as the "Heart of the Cards". Although he does not like Yugi and Jonouchi, he does not act outright hostile towards them, except during the first episode. However in the second season, his acceptance of faith and magic vanishes and in the English anime he adopts the habit of muttering that it's "hocus-pocus nonsense". His desire to defeat Yugi also grows from a grudge to an obsession and Kaiba shows an outright disregard for the safety of Yugi and his friends.
Despite this, he still often helps them with defeating the villains they face usually claiming that Yugi or one of his friends helped him and he thus owes them or that the villain's desires are in direct conflict with his own interests.
Deck & Strategy
Blue-Eyes White Dragon
Kaiba had "Duel Monsters" cards when he was a child, but his deck was very different than the one he uses later on. In fact, his signature monster (the three "Blue-Eyes White Dragon" cards), were his dream since he was a child. He wanted and liked that card so much that Mokuba decided to draw him one. He, of course, never used it in a duel but it remains as a sign of his little brother's love. Kaiba's love for the "Blue-Eyes White Dragon" card is probably what leads him to rip apart the fourth and only copy of the Blue-Eyes that wasn't in his deck. This fourth card belonged to Yugi (his grandfather actually) and Kaiba already had three Blue-Eyes White Dragons in his deck (three copies of the same card are the limit for any deck), all this factored into Kaiba's decision to rid Yugi of the fourth copy. He did it to weaken Yugi's deck and, probably out of some sort of spite, not wanting anyone else but him to have the Blue-Eyes in their deck. He was defeated anyway due to Yugi's "Exodia the Forbidden One" monster card.
The Blue-Eyes White Dragon remains his favorite and his trademark even today. He also has some different Blue-Eyes cards (like the ritual monster card, "White Dragon Paladin") and other unique cards (like "Kaibaman").
Kaiba, at one time, had the Egyptian God card known as "Obelisk the Tormentor". It was given to him by Ishizu Ishtar. While in the search for the power of the other Egyptian God cards, she had hoped he would organize a tournament that would relieve her brother, Marik, of his two Egyptian God cards (which indeed happened, in the end). Obelisk is the most powerful card that ever found its way into Kaiba's deck. He used it in many strategies and gained many victories in the short time that Obelisk was in his possession. He ultimately lost Obelisk to Yugi in the semi-finals of the very tournament he organized to gain the last two God cards in known existence. Yugi emerged from that tournament with all three Egyptian God cards in his deck.
Another powerful card he had was "The Fang of Critias" and later "Critias Knight". This was a powerful card, originating from Atlantis. It had an ability unique to the Divine Dragons ("The Eye of Timeus", "The Fang of Critias" and "The Claw of Helmos"), cards that only those chosen and are pure of heart receive and are able to use, and only to destroy the power of "The Seal of Oricalcos". The ability was to merge with other cards making them stronger (like "Polymerization", only without the actual card). He once used it on Yugi's "Mirror Force", which Yugi then used on a Dragon, creating a Mirror Dragon with an impressively powerful effect. He also used it with "Despotskaf Wing" on his "Blue-Eyes White Dragon", creating a "Blue-Eyes Despotskaf Dragon" of greater power. "The Fang of Critias" card was later changed into "Critias Knight" to allow a Fusion with "Timeus Knight" and "Helmos Knight" into a powerful Knight of Atlantis, using his effect (and defying logic) to defeat the evil Leviathan's infinite ATK and free the God cards it had consumed. The card has disappeared after Leviathan was destroyed.
In search of a way to defeat his bitter rival Yugi Moto and his God cards, Kaiba seeks help from Pegasus, the Creator of Games. He offers his three Blue-Eyes White Dragons as a stake in a duel, while Pegasus stakes one of his rare cards, the one that can defeat the God cards. Kaiba defeats Pegasus with surprising ease and takes two cards from Pegasus' batch, stating that he is a snake for telling him he only had one card to beat the Gods. Pegasus replies that there was only one, but Kaiba doesn't pay any attention. In truth, Anubis, the Egyptian Lord of the Dead, placed one of those two cards for Kaiba to take, so that by activating it, he releases Anubis from his slumber. That card was called "The Pyramid of Light", its apparent effect being to remove all current God cards from play. His plan was to force Yugi to summon all of his Gods at once, remove them from play, destroy the Pyramid card (using the other card he took from Pegasus), bring back the gods using "Return From the Different Dimension" and defeat Yugi using his own God cards. Anubis, however, doesn't like this and he activates the Pyramid's other effect, summoning "Andro Sphinx" and "Sphinx Teleia" and merging them into "Thenien the Ultimate Sphinx". Thenien consumes the souls of fallen monsters, amassing an ATK of 35000. Yugi destroys the Sphinx with the other card taken from Pegasus by Kaiba, the "Blue-Eyes Shining Dragon". This Dragon can only be summoned by offering "Blue-Eyes Ultimate Dragon" as a tribute. Although Ultimate is stronger (ATK 4500 DEF 3800) than Shining (ATK 3000 DEF 3000), Shining makes up for it with his effects. His ATK and DEF are increased by 300 for every Dragon Monster in the graveyard, which means he automatically has ATK and DEF of 4200 (Blue-Eyes Ultimate and the three regular Blue-Eyes used to summon the Ultimate). His other effect is called "Shining Nova" and it destroys the Shining Dragon and another card on the field. It is also immune to any Spells, Traps and monster effects, making it fairly difficult to destroy. After the events in "Yu-Gi-Oh! the Movie: the Pyramid of Light", the Pyramid card shatters, and the Shining Dragon is still, officially, in Kaiba's possession. The reason he doesn't use it in duels is unknown.
Kaiba's deck is composed mostly of Dragon type monsters and spells and traps that support them. He doesn't use exclusively Dragons though, and will use other powerful cards. Being an intelligent individual, he often uses different powerful combinations of cards (some of them are painfully simple, yet brutally effective) and is willing to make sacrifices to further his cause. He is a highly intelligent and skillful duelist with a deck and a wit to match. His only real weaknesses are his overconfidence and lack of patience when he's losing.
Maximillion Pegasus, the creator of games (Duel Monsters which he created), decides to hold a tournament on his private island and decides to call it Duel Kingdom. The cause for this tournament is supposedly to find a new King of Games, but in truth, Pegasus wants to collect souls for a dark spell to bring his dead wife back to life. The tournament system was the following: all contestants arrive by a (very large) ship, and they are given a duel gauntlet and a duel bracer. The bracer has 10 slots for 10 star tokens. Star tokens are won by winning duels and one needs 10 to participate in the finals. If one loses all tokens, he is out and goes home by the first boat. As if the cheating dorks who try to steal the tokens (rather than earn them) aren't enough, Pegasus allows "Eliminators" loose on the island. They have larger bracers, and are usually around both their wrists that can hold 20 or even 30 star tokens. They use different tactics to frighten, confuse or otherwise hamper opponents mentally before dueling them to increase their chances at victory as much as possible. The most well known are the Paradox brothers and Panik. When four people collect 10 tokens, they go into the castle and fight in a tournament-like duel system (they are set into groups of two and they fight the person from their group, the winners face off and the winner gets the reward) and the winner gets money and the chance to duel Pegasus for the title of the Master of Games.
Kaiba, however bent the rules a bit. Pegasus hacked into Kaiba Corp's system and started moving stocks to his name, slowly taking over the company. Kaiba, naturally, didn't quite like this, but when Pegasus kidnapped Mokuba, Kaiba's little brother, Kaiba decided to join the tournament and "teach Pegasus a thing or two". Right before the tournament, Kaiba fought Yugi and was defeated by his "Exodia the Forbidden One". He was hoping to meet Yugi again in the tournament to take his revenge. He defeated all competition easily (including Yugi's best friend, Joey Wheeler) and waited for Yugi at the entrance to Pegasus' castle. When Yugi came with his 10 star tokens, Kaiba appeared and gave him one of the two prototypes of the duel disc, so they could duel (Pegasus' island had special fields called "Duel Arenas" for fighting and duelists couldn't duel without one...until now). Yugi and Kaiba both put the discs on, and the duel begins. Kaiba, better prepared this time, faces Yugi (Kaiba doesn't know, however, that Yugi's Exodia is at the bottom of the ocean). Kaiba, with his new powerful card the "Blue-Eyes Ultimate Dragon" (all three Blue-Eyes White Dragons (BEWD) fused together) dominates the game, until Yugi uses his weakest card "Kuriboh" to defend from the final attack of BEUD and uses "Poison Arrow" to decrease BEUD's ATK from 4500 to a measley 900. Yugi is about to finish Kaiba when he decides to use a dirty trick. He stands on the edge of the castle wall saying that if Yugi attacks, Kaiba could fall to his death. At first, Yami Yugi decides to attack, but Tea calls out to him which forces the essence of Yugi Muto to stop Yami Yugi. Mentally broken, Yugi surrenders the duel causing him to lose five star tokens and a way into the castle. Although Kaiba had over 10 star tokens, he decided to break into Pegasus' castle and hack into his network to get his stolen stocks back when the transfer was at 99%. He then challenges Pegasus to a duel, alone, neglecting the rules. Pegasus accepts, but switches to a shadow game so Kaiba has to give his soul to Pegasus if he should lose. The duel starts and Pegasus activates his "Toon World" card (the effects are greatly exaggerated in the anime) and even takes BEWD away from Kaiba and uses him to beat Kaiba fairly easily (mostly because he used his Millennium Item, the Millennium Eye, to see Kaiba's hand and set cards, anticipating and preventing his every strategy). Kaiba's soul is captured, along with Mokuba's and that of Yugi's Grandpa, into a special card. Pegasus is eventually defeated by Yugi, and Kaiba's soul is freed from the card (and despite his soul being captured, he still doesn't believe in magic and calls it simple trickery). Pegasus returns Kaiba Corp to its rightful owner, while Kaiba returns home and thinks of a way to defeat Yugi, now holder of the title King of Games.
The rules for this duel in this tournament are quite primitive. Players get only 2000 LP each and any monster can be summoned right away, regardless of level (except fusions and rituals). And many card effects are exaggerated and are unrealistic, unusable or simply cannot be applied in the real game.