The body is a part vital to the complete existence of any living thing.
The Body gives the heart and body 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 body/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.
It should be noted that the body is one of the three things that make up existence of a living being. The other two are the 'heart' and the 'soul'.