A Branded Short Domain (BSD) is a custom domain that takes the place of bit.ly or bitly.com in a shortened URL. With a custom domain, the links you shorten in Bitly can be branded to increase brand recognition and clicks. For example, your link could be "mybrand.co/bigsale" instead of "bit.ly/bigsale".
You will need to purchase a custom domain from a third party domain registrar, like GoDaddy, Domai.nr, 101domain, or iwantmyname. To keep your links short, the custom domain you purchase must be 15 characters or less, including the dot.