Everyone has a Heart, Soul, and Body. Well, every normal person. Individual worlds also have Hearts, and in-game text alludes to them being somewhat sentient. Memory is also part of the equation, hinted to be another aspect of the Heart.

The Body is Exactly What It Says on the Tin—the thing that holds the Heart and is animated by the Soul.
The Heart contains the emotions and identity of an individual, and are made up of light and darkness. If the Heart is overcome by Darkness, it can separate itself from the Body and Soul and become a Heartless. It's accepted by most characters that a person without a Heart isn't really a person at all. The reality is slightly less complex; anyone or anything that wants to think of itself as a person can likely generate a Heart.
The Soul is seemingly simple—it's the animating force that gives life and movement to a body and heart. It's also translated as "Spirit" and synonymous with "Will" later in the series. Theoretically, if you somehow give or transplant a soul to an object, that object will gain a mind of its own and move around. In extreme circumstances, the Soul by itself could be enough to keep a Body moving. When a Heartless is formed, the Body and Soul left behind may reanimate into a Nobody, if the individual was particularly strong-willed.
Memory is hinted to be similar to one's Mind, and is critical to the function of a Heart, but not entirely the same. Nobodies have no Hearts but retain Memory to teach them how to fake emotions, Replicas are Artificial Humans made of Memory, and Memory can be used to strengthen or weaken one's Heart. Additionally, Chain of Memories says memories of different people can be connected, like links in a chain, and that chain can be broken, mended or reorganized. This means that as Sora forgets people as he proceeds through Castle Oblivion, the people he forgets about forget him, too.