A wiki is a web site that allows users to create and edit pages in an easy-to-format way (not HTML). They can easily create links on those pages to other pages in the wiki (and to pages on other web sites). The wiki usually keeps history of page edits, and allows rollback of pages to previous versions. New users can usually create accounts for themselves, and therefore page edits can be tracked based on user. Wikipedia is undoubtedly the most famous example of a wiki.
But what is a semantic wiki? I believe that there are four basic features that, when taken together, transform a wiki into a semantic wiki. While there can certainly be more features than just these four in a semantic wiki, I think that it is these four that must minimally be there to make a wiki semantic.
The first feature is that pages can be typed. That is, they can be marked as representing a certain "type" of thing, e.g. a book or a person or a city or an event. (Another word for type could be "category" or "class".) This type can simply be a word that has meaning to the wiki users, e.g. "Person", "City", etc. Different wiki technologies can differ on how this type is associated with a page (e.g., it can simply be another markup element that can be added to the wiki text of a page).
The second is that page links are assigned meaning. That is, hyperlinks from one page to another can be assigned more meaning than just "this is a link to a page"; the link can be assigned a "type". E.g. a link from a page about a book to a page about the author of that book might be assigned a type "authored-by", or "has-author". Again, this link type can simply be a word that has meaning to the wiki users, e.g. "authoredBy", "located-in", etc.
The third is that data values within a page can be assigned a meaning, e.g. the number 200,000 in a page about a city could be assigned the meaning of "population". Once again, "meaning", at its simplest level, is just associating a word from some vocabulary with the value. This can be thought of as an "attribute name" that goes with the value.

In summary, the four key elements that I believe make a wiki semantic are:
- The ability to type pages
- The ability to assign meaning to links between pages
- The ability to assign meaning to data values within a page, and
- The ability to query this knowledge to dynamically generate content
While there can certainly be more features than just these four in a semantic wiki, I think that it is these four that must minimally be there to make a wiki semantic.