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A domain you register for the sole purpose of having your own domain so you can have an easily remembered URL and e-mail address. The domain is usually served (often vhosted) off someone else's machines.
This is as opposed to a domain you register because you have machines of your own which are already on the Internet and which you want to make addressable via something other than dot addresses.
Whereas vanity domains were almost unheard-of in 1980s, since the invention and popularisation of the Web in the mid-1990s and the desire for URLs which consist only of memorable domain names (e.g., "http://pbs.org") for everything from movies to car wax, vanity domains have come to be the rule instead of the exception.