A franchise is a right granted by a government or corporation to an individual or group of individuals. One of the most important government-issued rights — so important it's known simply as "the franchise" — is the right to vote.
In its more commercial meaning, the fast-food chain McDonalds is a franchise. So are Burger King and Kentucky Fried Chicken and the New York Yankees. In other words, businesses from which you can buy a license in order to sell or use their products. More recently, the term has come to have an intellectual meaning as well, as in a particular series of books or films, such as the Harry Potter franchise.