Help:Purge
Purge is sometimes necessary to update a wiki page, when templates or subpages transclusion is involved. Purge clears the page's server cache, and the page is rebuilt. Before purging, you may want to try first to refresh the page using your web browser.
For updating a page display, any purge methods do the job, but for categories and backlinks a null edit explained below is required, and other methods don't work. Update of images is explained in a section below.
The page to purge is the one that transcludes, not the page that gets transcluded. Purge consumes a small amount of additional processing power to rebuild the page.
Theory
When a change is made to a page, the MediaWiki software saves the change to the main database. Thereafter a copy of that page is made to a "server cache" on secondary servers, and until the page is changed again, that cached page will be shown. A change in a subpage or template does not necessarily generate a new cache of the transcluder page for efficiency. A purge throws away the cache, and rebuilds the page again.
For images
To refresh thumbnails, you may have to purge either or both of the image page and/or the page that loads the image.
First go to the image description page by clicking the image. You should go further to Commons if the image is from Commons. Then purge. (Purge gadgets seem to work consistently. See commons:Help:Purge for Commons gadgets.) It may take a while for MediaWiki to complete rescaling, especially for an animated GIF. It depends on how busy the MediaWiki software is.
After a while, go back to the page with thumbnails and reload, bypassing your browser cache (Ctrl-F5 on Firefox). If the page still shows the old thumbnails, then purge this page.
If an image thumbnail will not yet regenerate successfully, it is probably because the server has cached an error page with the URL of a thumbnail, but the page is not purged because the actual thumbnail does not exist. The solution is to request the thumbnail with an unusual URL, say by appending "?1" to the end, then do a usual purge. If there is still a problem, ask for help at Wikipedia:Village pump (technical).