Heart
In the universe of Kingdom Hearts, the heart is a part vital to the complete existence of any living creature. Just as people have hearts, so do all worlds. Any being of a certain quality will averagely have a heart, a Body, and a Soul.
Hearts apparently give the soul and body emotion and feelings, as well as a conscience. Nobodies, who lack hearts, cannot feel or show emotions.
All hearts, with a couple exceptions, have both light and darkness within themselves. Such exceptions include the Princesses of Heart, who lack darkness and contain only pure light in their hearts. The light and darkness within a heart conflict and define a person, allowing them to exist. This is why nobodies are considered "not to exist"; because not only do they not physically exist, but because they lack the tool (the heart) that defines people and allows them to be.
If an individual, be they good or evil, gives into the darkness of their heart, they will become a Heartless, a being of pure darkness who seek out other pure hearts. This being separates from the body and soul to act on its own instinct. This primal instinct that one could consider to be reproduction drives the Heartless to seek other hearts and consume them, making more Heartless. However, when Heartless are defeated properly, the darkness releases its grip, and the captive heart is freed. Eventually, these freed hearts congregate and collect together to make Kingdom Hearts.
If a person with a strong heart becomes a Heartless, then the body and soul left behind may begin to act with a will of its own. This being is called a Nobody. If the person's heart was particularly strong, that Nobody may even retain its human form and memories, but will still lack true emotions which come from the heart. These specific nobodies are known as part of Organization XIII.
Memories are also tied with the heart. If a person's memory collapses, such as the Riku Replica, their heart will collapse as well, rendering their body into a lifeless doll. It is therefore safe to assume that, since the members of Organization XIII can remember certain things (such as how to break a heart), then they must have either shadow hearts, which contain memory, or fractions of their old hearts.
Connection to the Keyblade
It is said that anyone with a powerful heart may wield a Keyblade, but if one forgets he/she ever wielded a keyblade, it is no longer possible to wield it. It should be noted that Roxas is capable of wielding Keyblades, but this is due to him being the Nobody of Sora (who wields the keyblade) and indirectly inheriting his ability to wield.
In the first Kingdom Hearts game, there was only one keyblade. Near the end of the game, this Keyblade chose between Sora and Riku near based on whose heart was stronger. In Kingdom Hearts 2 there were multiple keyblades because the Realm of Inbetween is said to have chosen Riku as its Keyblade master, thus there is no longer a rivalry.
