To possess something is to have or own it. You can possess a physical object, you can possess a particular quality or skill, or you can possess control or influence over someone.
Related words are possessive (describing those who hold onto people or things in a controlling way), and possessed, as in a person being possessed by an idea or even by a demon. This verb is from Middle French posséder "to own," from Latin possidēre, from potis "able, having the power" plus sedēre "to sit."