The publish tool is just a shortcut for the publish functions that are built-in to several other web applications. It searches your websites and web applications and gives you links to publish each of them.
To publish the files that are associated with a website, click on the website's icon. Website icons look like folders.
To publish the files that are associated with a web application, click on the web application's icon. Most web applications do not have their own files, and will not be shown here. Some web applications will have uploaded files, however -- especially those that allow you to include images or documents. These web applications may have some built-in publishing functions to allow you to publish these files for better performance.
Published websites or applications will open a new window, listing files as they are published to disk. If any web pages are published, you may click on the web page filenames to visit the published version. Note that browser caching may still be saving previous versions of the published files. You may have to force a reload of the page to see the newly published version.