Depending on the specific wiki engine, the presence of PhpLanguage, MySql, etc. may be required. Many are implemented as Common Gateway Interface (CGI) programs, so you only need the web server. Of course, if the engine is written in a language that requires a runtime component, like PerlLanguage, PythonLanguage, or JavaLanguage, you'll need that environment as well.
Other engines are, indeed, written to use additional frameworks, like PHP and ZopeApplicationServer, and additional components like a database. Check the documentation of the particular engine for details.
See also WikiServer