Heart

The heart is a part vital to the complete existence of any living thing. Just as people have hearts, so do worlds.

Hearts appearently give the soul and body emotion and feelings, as well as a conscience. Nobodies, who lack hearts, cannot feel or show emotions.

All hearts have both light and darkness within them, except the Princesses of Heart and a few other, who lack darkness in their hearts.

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 hearts. Usually, the body turns into a Nobody. However, if the heart was strong, then the empty shell that is left behind will become a Nobody with a human form, and then seek to reunite with their hearts.

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 therefor safe to assume that, since the members of Organization XIII can remember things, such as how to break a heart, then they must have either shadow hearts, which contain memory, or fractions of their old hearts.

It should be noted that the 'heart' is one of the three things that make up existence of a living being. The other two are the 'body' and the 'soul.'

Connection to the Keyblade
It is said that anyone with a powerful heart may wield a Keyblade, but if the memories of wielding a Keyblade are forgotten, then it is no longer possible. It should be noted that Roxas is capable of wielding Keyblades, but this is due to him being the nobody of Sora and indirectly inheriting his ability to wield.