Wiki is a way for many people to collaborate together on writing one or more pages (in Wagn we call them cards), and flexibly link them together. No structure is enforced, but participants can develop some structure as they like collectively. The most famous example is Wikipedia.org, an encyclopedia that anyone can contribute to. There are many other public wikis on more particular subjects, and they are also used by groups to think together, collectively record information, develop documentation, organize events, etc.
You can think of a wiki as a book, with dozens to many thousands of pages and potentially infinite more. Each page has a name, and the name of a page is appropriate for the content. Off-topic content can be moved to a new page with an appropriate name. References among pages are simple, usually done by putting double brackets around the Page One Wants To Link To (or on some wikis, by SmushingCapitalizedWordsTogether).
The content of the pages is limited only by the imagination of the participants, because everyone can edit everything, and Adding New Pages is easy.
Wiki is a shortening of WikiWikiWeb, the name that Ward Cunningham gave to this type of web site when he created it in 1995 (see http://c2.com/cgi/wiki?WikiHistory ), to invite people to develop a Pattern language of programming. The name comes from the Hawaiian word wikiwiki, which translates to English as "fast" or "quick." His intent was to offer the simplest online database that could work.