You have a spirit, demon, or other spiritual being inhabiting your body. This spirit is usually opposed to you (but this does not always mean it is aggressive/harmful), seeking to make life inconvenient. Some spirits use your body to fulfill goals they wanted to fulfill in their past life, others possess the living for the thrill and nothing more.
When someone is using a weapon which only hits ghosts/spirits/incorporeal creatures, you are able to be hit by it as if you were incorporeal. Your spirit possessing you is unharmed, using your body's life force instead to ensure its own survival. In order to attack the spirit separate from you, you must receive an exorcism, and thus lose the Possessed trait.
The spirit possessing you has its own independent stats and ability scores, but can use your physical ability scores if it manages to take control of you and perform physical tasks. It does add a bonus when attempting these tasks at the expense of your Hit Points (the bonus is equal to double the Hit Points spent).