The body is a part vital to the complete existence of any living thing. It gives the heart and soul a vessel, granting a person a physical appearance and a form.
When someone becomes a Heartless, they leave behind their body and their soul, a shell. However, these shells become Nobodies if that person's heart was strong enough.
It should be noted that the empty shell that becomes a Nobody is actually a combination of the body and the soul. It is presumed that when a person dies, it is the body without the soul.
It also should be noted that the body is one of the three things that make up the existence of a living being. The other two are the 'heart' and the 'soul'.
A body can become complete on one of two conditions. The first is to meet up with the heart that has created a new body and willingly join with it. The other way is to call forth Kingdom Hearts, the entity of all living things, that would have the power to give a body back it's heart. This was the goal of Organization XIII for the entire length of Kingdom Hearts II.