Kingdom Hearts is one of the major objects in the Kingdom Hearts universe. It is the heart of all worlds, and the source of great power and wisdom. It can be seen as the heart of everything.
Kingdom Hearts
In the first game, Xehanort's Heartless and the horde villains led by Maleficent, seek out Kingdom Hearts. Both want to draw out the darkness that surrounded it, in order to gain immeasurable power and knowledge.
To accomplish this, Maleficent searches for the Seven Princesses of Heart. These maidens possessed hearts that had absolutely no darkness within them. When brought together, these princesses would reveal the final keyhole and the way to Kingdom Hearts itself. Thus, Maleficent attacks many worlds with the Heartless in her pursuit of them, and manages to capture all seven.
All the while, she is being used by Xehanort, who had been the first to discover the nature of hearts as well as Kingdom Hearts itself.
At Hollow Bastion, Xehanort's old residence, the seven princesses are finally gathered, but only when Sora sacrifices himself to release Kairi's heart, using a Keyblade constructed from the hearts of the six other maidens, is the path to Kingdom Hearts finally opened. With Xehanort having gained possession of Riku's body, forcing Riku's heart to wander into the darkness, he disappears behind the darkness that poured out from the keyhole, on the way to Kingdom Hearts.
After Sora is restored from his Heartless self, he once again goes to Hollow Bastion, and seals the Keyhole made by the princesses, holding off the great darkness. Regardless, Xehanort remains a threat, so Sora and co. venture to the End of the World, a place created out of the remains of the worlds taken by the Heartless. There, the Heartless thrive, and beyond it, in the eternal abyss, lies the door (now unlocked) to Kingdom Hearts.
After a long battle, Xehanort, wounded, reaches the endless abyss, where the Door to Darkness lies, with the Realm of Darkness and Kingdom Hearts waiting behind it. Xehanort forces the door to open, hoping for Kingdom Hearts to fill him with the power of darkness. However, to his horror, Kingdom Hearts is revealed not to be darkness, but light. The light overwhelms him, destroying him. But even with his defeat, the darkness that surrounds Kingdom Hearts threatens to overwhelm the Realm of Light. Because of this, Sora is forced to close the Door to Darkness, trapping Riku and King Mickey on the other side, in order to stop the Heartless threat. Nonetheless, the sealing also restores the many worlds, along with the barriers between them, while preventing billions of Heartless from invading the realm of light.
Kingdom Hearts II
Kingdom Hearts is again the goal of Xehanort in Kingdom Hearts II, but this time, it's his Nobody, Xemnas, and the Organization he leads. Maleficent, revived, also seeks out Kingdom Hearts once more.
However, Organization XIII wants Kingdom Hearts for an entirely different reason: they want to use its power in order to become whole, for they lead an incomplete existence as Nobodies. To fulfill their goal, Xemnas uses Sora (since Roxas, their original catalyst for releasing hearts, was no longer an independent entity), to destroy Heartless. The hearts that are released by Sora's Keyblade (and presumably all other Keyblades) would then wander to Kingdom Hearts, which floats over the Nobody world, The World That Never Was, as a heart-shaped moon, assimilating into it. When it is finished, the Organization would be able to become complete once more.
This plan is nearly realized until Ansem the Wise attempts to seal Kingdom Hearts away with a digital encoder. However, the machine is unable to process something as powerful and unpredictable as Kingdom Hearts, and it explodes, ostensibly killing Ansem. Still, this explosion leaves a giant hole in Kingdom Hearts, causing countless hearts to rain down over the world, becoming Heartless as soon as they touch the darkness. With Kingdom Hearts damaged, Xemnas needs to start all over again, though the threat of Sora forces him to take drastic action.
In order to gain the power to defeat Sora, Xemnas enters Kingdom Hearts, taking its power. Thankfully, a new door, much like the Door to Darkness, appears, allowing Sora and his companions to enter Kingdom Hearts as well. But Xemnas had already taken Kingdom Hearts' power, and attempts to use it against his opponents. Though he tries his best, and nearly succeeds, the combined efforts of Sora and his friends bring about his downfall. What ultimately happens to Kingdom Hearts after Xemnas' defeat is unknown.
Possible Connection to Birth By Sleep
In Birth By Sleep, the secret ending of Kingdom Hearts II: Final Mix, Master Xehanort, portrayed as an unknown villain in the video, conjures a blue, heart-shaped glowing object, identical to Kingdom Hearts except colored blue. The old man seems triumphant at having conjured the object, having summoned it with dark energy. The connection of this object to Kingdom Hearts remains unexplained.