Someone who's possessed is completely controlled by some force. If you've ever seen a crazed sports fan screaming with joy for the winning team, you've seen a person possessed with team spirit.
While possessed is the past tense of the verb possess, to have, it most often describes someone who's obsessed with something. The controlling force can be an emotion, like when your friend is possessed by her infatuation with her favorite pop star. It can also be a supernatural power or demon — possessed describes someone you believe is truly controlled by an evil spirit, or someone who just acts that way.