In KH, there are three important parts to life: the body, the heart, and the soul. Then when darkness takes over a person, new beings are formed: the heartless, nobodies, and now we see another type of entity being formed, the unbirths.

Heartless correspond with the person's heart.
Nobodies correspond with the person's body (I think)
So, I think that maybe Unbirths correspond with the person's soul.

That's just my guess on what Unbirths are, the person's soul after it has been corrupted by darkness.