A custom domain, sometimes called a vanity URL, is a way to create links that include your brand.
When a link includes your brand, it helps people recognize and associate your links with your product or company. When they see your brand, they trust your links, and that results in higher click-through rates.
For example, when ESPN shortens a link through Bitly, instead of being branded with Bitly's domain (bit.ly/wG8qhgpn) the link displays their own custom domain (es.pn/2Qw1wd7).
Custom domains + Bitly
To use a custom domain in Bitly, it must be no more than 32 characters long, including the dot.
Purchasing or adding a custom domain in your Bitly account replaces the bit.ly in your short links with your own custom domain. In the ESPN example above, es.pn replaces bit.ly.
By editing the end of the URL - we call that the back-half - you can further customize the link to be an even more identifiable Custom Branded Link, like es.pn/superbowl. Read more about that in What is a Custom Branded Link?
Please note this domain cannot be the same exact domain that points to your website, but you can direct it there. See How do I redirect visitors from my custom domain or an unknown URL?