What are the applications of brand?
Magic has this definition for "permanent":


  110.1. A permanent is a card or token on the battlefield. A permanent remains on the battlefield indefinitely. A card or token becomes a permanent as it enters the battlefield and it stops being a permanent as it’s moved to another zone by an effect or rule.


So, Brand only cares about cards and tokens currently on the battlefield.

Magic has the following rules about ownership:


  
  108.3. The owner of a card in the game is the player who started the game with it in his or her deck. If a card is brought into the game from outside the game rather than starting in a player’s deck, its owner is the player who brought it into the game. If a card starts the game in the command zone, its owner is the player who put it into the command zone to start the game. Legal ownership of a card in the game is irrelevant to the game rules except for the rules for ante.
  110.5a A token is both owned and controlled by the player under whose control it entered the battlefield.
  


And one about control:


  
  110.2. A permanent’s owner is the same as the owner of the card that represents it (unless it’s a token; see rule 110.5a). A permanent’s controller is, by default, the player under whose control it entered the battlefield. Every permanent has a controller.
  


Basically, the permanents you own are the cards you put in your deck and the tokens you made, and you start out controlling the permanents that you put into play.

There are some cards that allow you to put cards your opponents own into play, like Beacon of Unrest, Bribery, and Gather Specimens.

Other cards let you take control of a permanent already on the battlefield, such as Act of Treason and Blatant Thievery.

Still others allow you to exchange control of a permanent you control and a permanent someone else controls, such as Daring Thief and Shifting Loyalties.

Some cards even allow you to give an opponent control of a permanent you control, such as Donate and Zedruu the Greathearted.

Brand allows you to regain control of any permanents you lost control of in any of those ways. For example, you can exchange control of a bunch of permanents you control with permanents your opponent controls, and then cast Brand to get your permanents back.