The body is a part vital to the complete existence of any living thing.
The Body gives the heart and soul a vessel to live in, giving a person a physical appearance and a form. When someone becomes a Heartless, they leave behind their body and their soul, a shell. Usually, the soul fades away. These shells become Nobodies however, if that person's heart was strong enough. If the heart of a person is strong, their shell becomes strong also, in longing for a heart, with the body giving the Nobody form.
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.
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 (Special to Sora and Kairi) and willingly join to it. The other way is to call forth Kingdom Hearts, the entity of all living things, that would have the power to giva a body back it's heart. This was the goal of Organization XIII for the entire length of Kingdom Hearts II