To run your own copy of DokuWiki you will need the following:
DokuWiki will run on any Webserver supporting PHP. Most users use Apache but many others including IIS, lighttpd and Nginx are known to work.
PHP needs to be at least Version 5.1.2.
For resizing images either the PHP GD extension or Image Magick should be installed.
DokuWiki should work in PHP's Safe Mode, but depending on your hosting configuration you may need to use the safemodehack option.
Any browser of the last few years should work. Firefox should be used for best experience…