This tutorial is going to use Joomla! for it's example case, but this same technique can be used for any software package.

Step 1: Get A Diff Tool

There are many tools out there to recursively compare two directories.  On Windows, my tool of choice is WinMerge and can be downloaded here:

http://winmerge.org/

Step 2: Backup Your Site

Most web hosting file managers have a way to archive files on a site.  For this step, archive all the files on your site into a compressed archive file and download it locally.  You can also use FTP to download a copy of all the files on your site to your local machine.  There are also specific tools out there to help do this for different CMSs.  The end result we want is a duplicate copy of all the files on your website on your local machine.

Step 3: Obtain The Original Package

Go to the website for your software and download the original package you installed.  Make sure the version number matches the version number you are currently running on your web site.  We want to compare what you have with what you originally installed.  If you still have the package you originally downloaded, use it.

Step 4: Compare The Files

Once you have your backup directory and you have unzipped the original package, you can compare the two using WinMerge:

WinMerge will show you all the files that are different between your two directories.  Make sure to "Include Subfolders" to make sure all the files in your subdirectories are compared.  WinMerge will then show you the results of your comparison.  It is helpful to change which information is displayed in the View menu:

You can double click files to see the exact changes that were made to the files:

Using this technique, you can see what changes have been made to the files on your site and take the appropriate action.

Note: You may want to edit the Options in WinMerge and tell the tool to ignore Whitespace differences.  If you do not, a blank line, extra carriage return at the end of the file, or different newline characters will cause the files to look different when they will actually function the same.