Once a person claims a body, it means that they are somehow close to the dead person - either by blood, marriage and /or close friendship - and that they will take responsibility for what is done with the body. This can mean burial or cremation.

Someone is dead and the body is with the authorities -- at a morgue, a hospital, the coroner, etc. The dead person's nearest relative(s) have not come to have the body transported to a funeral home for burial or cremation.