A one-user wiki is a wiki that has only one active user. Sometimes this is the owner's intention, as when he configures his site in a ByInvitation format and then does not find anyone he wishes to invite. Other times, it is by accident, as when the owner sets up his wiki as OpenEdit but no one wants to edit.
The use of wiki software in a site with only one user is sometimes deemed pointless, since the software is designed for collaboration. There are many other content management systems that are geared more toward maintenance by one user.
However, the use of wiki software, particularly MediaWiki, for a one-user site does have some advantages. If he already has familiarity with MediaWiki, then using it for his site may help him avoid the learning curve that would be involved with a different CMS. Also, the content can easily be exported to other wikis. There is also the theoretical possibility that eventually, more users will join the site (although in some cases this would require switching to OpenEdit or a similar format).
Some one-user wikis have created more content than wikis with several users, because the one user was more prolific than those other users combined. Having only one user also eliminates the overhead of creating a bunch of project namespace pages for coordinating and governing user activity. The user also is freed from monitoring other users' activity, and trying to reach consensus with them, to focus on his own work.