A lot of people are surprised we don't just have a button to delete a wiki article.  While that would be nice, it's not that simple.  Wiki articles are usually linked to other wiki articles, and these inbound links need to be dealt with before we can move forward with deletion.

Simply go to Wiki > Customize and choose your wiki to edit.  If on the Edit Wiki page you click the "List all articles in this Wiki" link, you'll be taken to a list of all the articles, each of them with a count of inbound links.  Click the number of inbound links and you'll get a pop-up listing the articles that link to the article you'd like to delete.  Once you've edited those articles to remove the links, you can go back to the Edit Wiki page and select "List all orphaned articles in this Wiki".  Your article should now appear in this list, along with a handy delete button.