A vanity URL is a branded and descriptive URL within a domain or website where URLs are usually random, generic and non significant.
The distinction between vanity and generic URLs is mainly met for social networking sites which often offer generic URL by default.
www.facebook/dietcoke is for instance a vanity URL and www.facebook/u21grd56 a generic one.
On Vine website a Vine vanity URL will appear as "vine.co/(username)/".
A vanity URL has better visual impact and is more significant in SERP. It also offers the ability to share profile pages more easily and makes it much easier for users to be discovered.
It is also better for memorization when used by brands in direct marketing or advertising on traditionnal media.