Who knows! :- Here's a few possibilities though.
(a)Your astral body. If the soul exists, then it could possibly be descibed as an astral representation of the pysical self, in a quite traditional classic art sense. The essense of what makes us alive. This description of the soul is one in which the soul can wander away from the body in the form of either astral projection, an OOBE or an NDE. You can actually exist as 2 people, but only experience things as your astral self. People who have had NDE's often claim to have hovered above their bodies, and seen a silver thread of energy connecting the solar plexus of their astral self, to the solar plexus of their physical body. In this concept of the soul, the body is simply a shell, all that is you is your soul. This is the type of soul related to described by those who believe in the afterlife, reincarnation, auras, or the astral body.
(b) Your individuality. A much more philosophical way of looking at the soul is that it is not in any way material. The soul is the part of you that stops you from simply being a mindless automaton. This type of soul is the part of you that makes all those rash illogical descisions, the part of you that analyses everything differently to everyone else, the part of you that makes you ... ... ... well you. In this concept of the soul, it cannot leave your body, as it is not at all physical. The soul is fully integrated with the body, once it leaves you, you die, what happens after is a much debated topic.
(c) Your empathy. The part of you that can feel other peoples emotion. Some believe that the soul is your emotional center, the part of us that makes us perceptive of each others pains, desires, etc. A psychic energy that is part of you, and linked spiritually to all other souls.
(d) Human arrogance! It is entirely possible of course that there is no soul, and that the arrogant human desire to see beyond the realms of the world around us has led our species to concoct the concept of the soul as an assurance that we have more than 80 years to live. The notion that we are simply born and die through a chemical process is quite a bleak one, the possibility that the soul does not exist should therefore be considered. It is in our nature as a species to try to deny the inevitable, therefore the possibility that the soul is simply an imaginary form of pyschological security that once we die we are not gone for good, should not be overlooked. After all very few people are truly prepared to simply not exist any more, the concept of an afterlife also helps grieving, the possibility that we just die and that's it, is too hard for some people to accept.
(e) A psychic shadow. As demonstrated be anyone who has ever taken and/or seen any Kirlian photography. A warm healing energy surrounding and engulfing the physical body.