Collaring, in BDSM, is a formal ceremony that establishes an ownership relationship. A submissive or slave is collared by their master or mistress making them officially theirs. This is done by ceremonially putting a collar around the sub's neck.
When a person is collared, it means their owner has exclusive say over them. A collared person is generally not available for play with other people, except by explicit permission (or command) from their owner.
Collaring sometimes takes the form of an elaborate ceremony, with friends and spectators from the BDSM community present to witness and celebrate the two, much like in a marriage ceremony. Before the collaring ceremony, sometimes a contract is formally signed by both parties that determines the details of their relationship.