Organization XIII is a group of thirteen powerful Nobodies who seek to reclaim their hearts and become whole again. They are the antagonists of Kingdom Hearts: Chain of Memories and Kingdom Hearts II, all will be playable in the upcoming game Kingdom Hearts 358/2 Days which happens after the ending events of Chain of Memories. There will also be a fourteeth member which is known to be female.
Organization XIII debuted in Kingdom Hearts: Chain of Memories, then known as simply, "the Organization." In Sora's story, the player was introduced to (in no particular order) Marluxia, Axel, Larxene, and Vexen. In Reverse/Rebirth, Zexion and Lexaeus were added to the cast. The Organization also appeared in Kingdom Hearts II. In this second installment, the player meets (once again, in no particular order) Xigbar, Xaldin, Saïx, Demyx, Luxord, Roxas, the leader Xemnas, and the returning member, Axel.
As they lack hearts, they have no emotions. Many of them try to hide this fact by pretending that they posses these qualities, fooling others as well as themselves. The Organization yearns to discover the secrets of the heart and fill the empty void inside themselves by gathering hearts to form Kingdom Hearts so they can become full beings again. While each member is aware of their lack of emotions - a fact that makes them feel inadequate - they fill their non-existence with their favorite pastimes. The Kingdom Hearts: Chain of Memories manga and several other media forms provide an alternative view, saying that they do have emotions, but are plagued by a feeling of uncertainty and emptiness due to not having hearts.
Their inclusion to the Kingdom Hearts storyline marks the more mature nature of the Chain of Memories and Kingdom Hearts II story lines as they introduce the concepts of duplicity among the villain, minor physical abuse, manipulation of sentimental matters and, on some level, real death (the Heartless invasion of Hallow Bastion which they caused inadvertently incapacitates Goofy, leading the player to falsely believe that he is dead). It is possible that there are heartless versions of the members of the Organization (with the exception of Xenmas who's heartless form was destroyed by Sora, Donald and Goofy in the events of the first game) whether this will be touched upon in a later game is to be seen.