In terms of BDSM, an owner is one who takes responsibility for, and has authority over, another person. The other person is usually called a slave. The slave will usually live with the owner but not always. This is a permanent relationship (and, if they live together, possibly 24/7), not related to a play scene, although people in a scene may like to act out a fantasy of being owner and property.
Ownership of one person by another is not legal and there is no implication that this is an attempt to create a legal position. It is a consensual arrangment entered into by people who understand themselves and how they want to relate to another person.
To be an owner requires a high level of dedication, personal awareness and responsibility. While the rewards can be great, it is something that few people are able to sustain.
Typically, an owner is referred to as Master (if male) or Mistress (if female) by their property, or slave. This is not universal, however, and there are Owners who prefer other titles or simply to have their name used. The slave is not always referred to as a slave.